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## G = C2×C21.C32order 378 = 2·33·7

### Direct product of C2 and C21.C32

Aliases: C2×C21.C32, C42.5C32, C1433- 1+2, C7⋊C96C6, (C3×C42).3C3, (C3×C21).8C6, C21.11(C3×C6), C76(C2×3- 1+2), (C2×C7⋊C9)⋊3C3, C6.5(C3×C7⋊C3), C3.5(C6×C7⋊C3), (C3×C6).(C7⋊C3), C32.(C2×C7⋊C3), SmallGroup(378,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×C21.C32
 Chief series C1 — C7 — C21 — C3×C21 — C21.C32 — C2×C21.C32
 Lower central C7 — C21 — C2×C21.C32
 Upper central C1 — C6 — C3×C6

Generators and relations for C2×C21.C32
G = < a,b,c,d | a2=b21=d3=1, c3=b7, ab=ba, ac=ca, ad=da, cbc-1=b4, bd=db, dcd-1=b7c >

Smallest permutation representation of C2×C21.C32
On 126 points
Generators in S126
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 106)(61 107)(62 108)(63 109)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)(22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(1 62 26 8 48 33 15 55 40)(2 57 30 9 43 37 16 50 23)(3 52 34 10 59 41 17 45 27)(4 47 38 11 54 24 18 61 31)(5 63 42 12 49 28 19 56 35)(6 58 25 13 44 32 20 51 39)(7 53 29 14 60 36 21 46 22)(64 116 102 71 123 88 78 109 95)(65 111 85 72 118 92 79 125 99)(66 106 89 73 113 96 80 120 103)(67 122 93 74 108 100 81 115 86)(68 117 97 75 124 104 82 110 90)(69 112 101 76 119 87 83 126 94)(70 107 105 77 114 91 84 121 98)
(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)(85 92 99)(86 93 100)(87 94 101)(88 95 102)(89 96 103)(90 97 104)(91 98 105)(106 120 113)(107 121 114)(108 122 115)(109 123 116)(110 124 117)(111 125 118)(112 126 119)

G:=sub<Sym(126)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,106)(61,107)(62,108)(63,109), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,62,26,8,48,33,15,55,40)(2,57,30,9,43,37,16,50,23)(3,52,34,10,59,41,17,45,27)(4,47,38,11,54,24,18,61,31)(5,63,42,12,49,28,19,56,35)(6,58,25,13,44,32,20,51,39)(7,53,29,14,60,36,21,46,22)(64,116,102,71,123,88,78,109,95)(65,111,85,72,118,92,79,125,99)(66,106,89,73,113,96,80,120,103)(67,122,93,74,108,100,81,115,86)(68,117,97,75,124,104,82,110,90)(69,112,101,76,119,87,83,126,94)(70,107,105,77,114,91,84,121,98), (22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119)>;

G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,106)(61,107)(62,108)(63,109), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,62,26,8,48,33,15,55,40)(2,57,30,9,43,37,16,50,23)(3,52,34,10,59,41,17,45,27)(4,47,38,11,54,24,18,61,31)(5,63,42,12,49,28,19,56,35)(6,58,25,13,44,32,20,51,39)(7,53,29,14,60,36,21,46,22)(64,116,102,71,123,88,78,109,95)(65,111,85,72,118,92,79,125,99)(66,106,89,73,113,96,80,120,103)(67,122,93,74,108,100,81,115,86)(68,117,97,75,124,104,82,110,90)(69,112,101,76,119,87,83,126,94)(70,107,105,77,114,91,84,121,98), (22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119) );

G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,106),(61,107),(62,108),(63,109)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21),(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(1,62,26,8,48,33,15,55,40),(2,57,30,9,43,37,16,50,23),(3,52,34,10,59,41,17,45,27),(4,47,38,11,54,24,18,61,31),(5,63,42,12,49,28,19,56,35),(6,58,25,13,44,32,20,51,39),(7,53,29,14,60,36,21,46,22),(64,116,102,71,123,88,78,109,95),(65,111,85,72,118,92,79,125,99),(66,106,89,73,113,96,80,120,103),(67,122,93,74,108,100,81,115,86),(68,117,97,75,124,104,82,110,90),(69,112,101,76,119,87,83,126,94),(70,107,105,77,114,91,84,121,98)], [(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56),(85,92,99),(86,93,100),(87,94,101),(88,95,102),(89,96,103),(90,97,104),(91,98,105),(106,120,113),(107,121,114),(108,122,115),(109,123,116),(110,124,117),(111,125,118),(112,126,119)]])

58 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 7A 7B 9A ··· 9F 14A 14B 18A ··· 18F 21A ··· 21P 42A ··· 42P order 1 2 3 3 3 3 6 6 6 6 7 7 9 ··· 9 14 14 18 ··· 18 21 ··· 21 42 ··· 42 size 1 1 1 1 3 3 1 1 3 3 3 3 21 ··· 21 3 3 21 ··· 21 3 ··· 3 3 ··· 3

58 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C7⋊C3 3- 1+2 C2×C7⋊C3 C2×3- 1+2 C3×C7⋊C3 C6×C7⋊C3 C21.C32 C2×C21.C32 kernel C2×C21.C32 C21.C32 C2×C7⋊C9 C3×C42 C7⋊C9 C3×C21 C3×C6 C14 C32 C7 C6 C3 C2 C1 # reps 1 1 6 2 6 2 2 2 2 2 4 4 12 12

Matrix representation of C2×C21.C32 in GL4(𝔽127) generated by

 126 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 61 0 0 0 0 87 0 0 0 0 47
,
 19 0 0 0 0 0 1 0 0 0 0 1 0 107 0 0
,
 19 0 0 0 0 1 0 0 0 0 107 0 0 0 0 19
G:=sub<GL(4,GF(127))| [126,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,61,0,0,0,0,87,0,0,0,0,47],[19,0,0,0,0,0,0,107,0,1,0,0,0,0,1,0],[19,0,0,0,0,1,0,0,0,0,107,0,0,0,0,19] >;

C2×C21.C32 in GAP, Magma, Sage, TeX

C_2\times C_{21}.C_3^2
% in TeX

G:=Group("C2xC21.C3^2");
// GroupNames label

G:=SmallGroup(378,27);
// by ID

G=gap.SmallGroup(378,27);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,96,187,1359]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^21=d^3=1,c^3=b^7,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^4,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

Export

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