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G = C2×C63⋊3C3order 378 = 2·33·7

Direct product of C2 and C63⋊3C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×C63⋊3C3
 Chief series C1 — C7 — C21 — C63 — C63⋊3C3 — C2×C63⋊3C3
 Lower central C7 — C21 — C2×C63⋊3C3
 Upper central C1 — C6 — C18

Generators and relations for C2×C633C3
G = < a,b,c | a2=b63=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C2×C633C3
On 126 points
Generators in S126
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 64)(60 65)(61 66)(62 67)(63 68)
(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)
(2 17 5)(3 33 9)(4 49 13)(6 18 21)(7 34 25)(8 50 29)(10 19 37)(11 35 41)(12 51 45)(14 20 53)(15 36 57)(16 52 61)(23 38 26)(24 54 30)(27 39 42)(28 55 46)(31 40 58)(32 56 62)(44 59 47)(48 60 63)(64 115 112)(65 68 116)(66 84 120)(67 100 124)(70 85 73)(71 101 77)(72 117 81)(74 86 89)(75 102 93)(76 118 97)(78 87 105)(79 103 109)(80 119 113)(82 88 121)(83 104 125)(91 106 94)(92 122 98)(95 107 110)(96 123 114)(99 108 126)

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

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

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

58 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 7A 7B 9A 9B 9C 9D 9E 9F 14A 14B 18A 18B 18C 18D 18E 18F 21A 21B 21C 21D 42A 42B 42C 42D 63A ··· 63L 126A ··· 126L order 1 2 3 3 3 3 6 6 6 6 7 7 9 9 9 9 9 9 14 14 18 18 18 18 18 18 21 21 21 21 42 42 42 42 63 ··· 63 126 ··· 126 size 1 1 1 1 21 21 1 1 21 21 3 3 3 3 21 21 21 21 3 3 3 3 21 21 21 21 3 3 3 3 3 3 3 3 3 ··· 3 3 ··· 3

58 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C7⋊C3 3- 1+2 C2×C7⋊C3 C2×3- 1+2 C3×C7⋊C3 C6×C7⋊C3 C63⋊3C3 C2×C63⋊3C3 kernel C2×C63⋊3C3 C63⋊3C3 C2×C7⋊C9 C126 C6×C7⋊C3 C7⋊C9 C63 C3×C7⋊C3 C18 C14 C9 C7 C6 C3 C2 C1 # reps 1 1 4 2 2 4 2 2 2 2 2 2 4 4 12 12

Matrix representation of C2×C633C3 in GL3(𝔽127) generated by

 126 0 0 0 126 0 0 0 126
,
 53 13 2 2 9 94 94 93 6
,
 1 0 0 104 126 126 0 1 0
G:=sub<GL(3,GF(127))| [126,0,0,0,126,0,0,0,126],[53,2,94,13,9,93,2,94,6],[1,104,0,0,126,1,0,126,0] >;

C2×C633C3 in GAP, Magma, Sage, TeX

C_2\times C_{63}\rtimes_3C_3
% in TeX

G:=Group("C2xC63:3C3");
// GroupNames label

G:=SmallGroup(378,25);
// by ID

G=gap.SmallGroup(378,25);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,187,102,1359]);
// Polycyclic

G:=Group<a,b,c|a^2=b^63=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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