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G = C2×C6.11D12order 288 = 25·32

Direct product of C2 and C6.11D12

direct product, metabelian, supersoluble, monomial

Aliases: C2×C6.11D12, C62.95D4, C62.248C23, C62(D6⋊C4), (C2×C12)⋊27D6, (C22×C12)⋊8S3, (C2×C6).43D12, C6.63(C2×D12), (C6×C12)⋊29C22, C62.87(C2×C4), (C22×C6).157D6, (C2×C62).109C22, C22.16(C12⋊S3), C22.20(C327D4), (C2×C6×C12)⋊3C2, C33(C2×D6⋊C4), C6.79(S3×C2×C4), (C22×C3⋊S3)⋊7C4, (C2×C6).57(C4×S3), C2.3(C2×C12⋊S3), (C3×C6)⋊6(C22⋊C4), (C22×C4)⋊3(C3⋊S3), (C3×C6).274(C2×D4), C6.115(C2×C3⋊D4), (C23×C3⋊S3).4C2, C22.17(C4×C3⋊S3), C23.37(C2×C3⋊S3), C3212(C2×C22⋊C4), C2.2(C2×C327D4), (C2×C6).96(C3⋊D4), (C22×C3⋊Dic3)⋊7C2, (C2×C6).265(C22×S3), (C3×C6).110(C22×C4), (C2×C3⋊Dic3)⋊18C22, C22.23(C22×C3⋊S3), (C22×C3⋊S3).90C22, (C2×C4)⋊8(C2×C3⋊S3), C2.19(C2×C4×C3⋊S3), (C2×C3⋊S3)⋊17(C2×C4), SmallGroup(288,784)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C6.11D12
C1C3C32C3×C6C62C22×C3⋊S3C23×C3⋊S3 — C2×C6.11D12
C32C3×C6 — C2×C6.11D12
C1C23C22×C4

Generators and relations for C2×C6.11D12
 G = < a,b,c,d | a2=b6=c12=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 1636 in 396 conjugacy classes, 133 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×C22⋊C4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C2×D6⋊C4, C6.11D12, C22×C3⋊Dic3, C2×C6×C12, C23×C3⋊S3, C2×C6.11D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C3⋊S3, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C2×C3⋊S3, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, C22×C3⋊S3, C2×D6⋊C4, C6.11D12, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C327D4, C2×C6.11D12

Smallest permutation representation of C2×C6.11D12
On 144 points
Generators in S144
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 80)(26 81)(27 82)(28 83)(29 84)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 97)(59 98)(60 99)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 121)(69 122)(70 123)(71 124)(72 125)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 133)(94 134)(95 135)(96 136)
(1 68 80 108 141 40)(2 69 81 97 142 41)(3 70 82 98 143 42)(4 71 83 99 144 43)(5 72 84 100 133 44)(6 61 73 101 134 45)(7 62 74 102 135 46)(8 63 75 103 136 47)(9 64 76 104 137 48)(10 65 77 105 138 37)(11 66 78 106 139 38)(12 67 79 107 140 39)(13 114 126 30 50 94)(14 115 127 31 51 95)(15 116 128 32 52 96)(16 117 129 33 53 85)(17 118 130 34 54 86)(18 119 131 35 55 87)(19 120 132 36 56 88)(20 109 121 25 57 89)(21 110 122 26 58 90)(22 111 123 27 59 91)(23 112 124 28 60 92)(24 113 125 29 49 93)
(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)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 107 108 12)(2 11 97 106)(3 105 98 10)(4 9 99 104)(5 103 100 8)(6 7 101 102)(13 127 30 95)(14 94 31 126)(15 125 32 93)(16 92 33 124)(17 123 34 91)(18 90 35 122)(19 121 36 89)(20 88 25 132)(21 131 26 87)(22 86 27 130)(23 129 28 85)(24 96 29 128)(37 70 77 143)(38 142 78 69)(39 68 79 141)(40 140 80 67)(41 66 81 139)(42 138 82 65)(43 64 83 137)(44 136 84 63)(45 62 73 135)(46 134 74 61)(47 72 75 133)(48 144 76 71)(49 116 113 52)(50 51 114 115)(53 112 117 60)(54 59 118 111)(55 110 119 58)(56 57 120 109)

G:=sub<Sym(144)| (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,80)(26,81)(27,82)(28,83)(29,84)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,97)(59,98)(60,99)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,121)(69,122)(70,123)(71,124)(72,125)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,133)(94,134)(95,135)(96,136), (1,68,80,108,141,40)(2,69,81,97,142,41)(3,70,82,98,143,42)(4,71,83,99,144,43)(5,72,84,100,133,44)(6,61,73,101,134,45)(7,62,74,102,135,46)(8,63,75,103,136,47)(9,64,76,104,137,48)(10,65,77,105,138,37)(11,66,78,106,139,38)(12,67,79,107,140,39)(13,114,126,30,50,94)(14,115,127,31,51,95)(15,116,128,32,52,96)(16,117,129,33,53,85)(17,118,130,34,54,86)(18,119,131,35,55,87)(19,120,132,36,56,88)(20,109,121,25,57,89)(21,110,122,26,58,90)(22,111,123,27,59,91)(23,112,124,28,60,92)(24,113,125,29,49,93), (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,107,108,12)(2,11,97,106)(3,105,98,10)(4,9,99,104)(5,103,100,8)(6,7,101,102)(13,127,30,95)(14,94,31,126)(15,125,32,93)(16,92,33,124)(17,123,34,91)(18,90,35,122)(19,121,36,89)(20,88,25,132)(21,131,26,87)(22,86,27,130)(23,129,28,85)(24,96,29,128)(37,70,77,143)(38,142,78,69)(39,68,79,141)(40,140,80,67)(41,66,81,139)(42,138,82,65)(43,64,83,137)(44,136,84,63)(45,62,73,135)(46,134,74,61)(47,72,75,133)(48,144,76,71)(49,116,113,52)(50,51,114,115)(53,112,117,60)(54,59,118,111)(55,110,119,58)(56,57,120,109)>;

G:=Group( (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,80)(26,81)(27,82)(28,83)(29,84)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,97)(59,98)(60,99)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,121)(69,122)(70,123)(71,124)(72,125)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,133)(94,134)(95,135)(96,136), (1,68,80,108,141,40)(2,69,81,97,142,41)(3,70,82,98,143,42)(4,71,83,99,144,43)(5,72,84,100,133,44)(6,61,73,101,134,45)(7,62,74,102,135,46)(8,63,75,103,136,47)(9,64,76,104,137,48)(10,65,77,105,138,37)(11,66,78,106,139,38)(12,67,79,107,140,39)(13,114,126,30,50,94)(14,115,127,31,51,95)(15,116,128,32,52,96)(16,117,129,33,53,85)(17,118,130,34,54,86)(18,119,131,35,55,87)(19,120,132,36,56,88)(20,109,121,25,57,89)(21,110,122,26,58,90)(22,111,123,27,59,91)(23,112,124,28,60,92)(24,113,125,29,49,93), (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,107,108,12)(2,11,97,106)(3,105,98,10)(4,9,99,104)(5,103,100,8)(6,7,101,102)(13,127,30,95)(14,94,31,126)(15,125,32,93)(16,92,33,124)(17,123,34,91)(18,90,35,122)(19,121,36,89)(20,88,25,132)(21,131,26,87)(22,86,27,130)(23,129,28,85)(24,96,29,128)(37,70,77,143)(38,142,78,69)(39,68,79,141)(40,140,80,67)(41,66,81,139)(42,138,82,65)(43,64,83,137)(44,136,84,63)(45,62,73,135)(46,134,74,61)(47,72,75,133)(48,144,76,71)(49,116,113,52)(50,51,114,115)(53,112,117,60)(54,59,118,111)(55,110,119,58)(56,57,120,109) );

G=PermutationGroup([[(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,115),(8,116),(9,117),(10,118),(11,119),(12,120),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,80),(26,81),(27,82),(28,83),(29,84),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,97),(59,98),(60,99),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,121),(69,122),(70,123),(71,124),(72,125),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,133),(94,134),(95,135),(96,136)], [(1,68,80,108,141,40),(2,69,81,97,142,41),(3,70,82,98,143,42),(4,71,83,99,144,43),(5,72,84,100,133,44),(6,61,73,101,134,45),(7,62,74,102,135,46),(8,63,75,103,136,47),(9,64,76,104,137,48),(10,65,77,105,138,37),(11,66,78,106,139,38),(12,67,79,107,140,39),(13,114,126,30,50,94),(14,115,127,31,51,95),(15,116,128,32,52,96),(16,117,129,33,53,85),(17,118,130,34,54,86),(18,119,131,35,55,87),(19,120,132,36,56,88),(20,109,121,25,57,89),(21,110,122,26,58,90),(22,111,123,27,59,91),(23,112,124,28,60,92),(24,113,125,29,49,93)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,107,108,12),(2,11,97,106),(3,105,98,10),(4,9,99,104),(5,103,100,8),(6,7,101,102),(13,127,30,95),(14,94,31,126),(15,125,32,93),(16,92,33,124),(17,123,34,91),(18,90,35,122),(19,121,36,89),(20,88,25,132),(21,131,26,87),(22,86,27,130),(23,129,28,85),(24,96,29,128),(37,70,77,143),(38,142,78,69),(39,68,79,141),(40,140,80,67),(41,66,81,139),(42,138,82,65),(43,64,83,137),(44,136,84,63),(45,62,73,135),(46,134,74,61),(47,72,75,133),(48,144,76,71),(49,116,113,52),(50,51,114,115),(53,112,117,60),(54,59,118,111),(55,110,119,58),(56,57,120,109)]])

84 conjugacy classes

class 1 2A···2G2H2I2J2K3A3B3C3D4A4B4C4D4E4F4G4H6A···6AB12A···12AF
order12···222223333444444446···612···12
size11···11818181822222222181818182···22···2

84 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C4S3D4D6D6C4×S3D12C3⋊D4
kernelC2×C6.11D12C6.11D12C22×C3⋊Dic3C2×C6×C12C23×C3⋊S3C22×C3⋊S3C22×C12C62C2×C12C22×C6C2×C6C2×C6C2×C6
# reps1411184484161616

Matrix representation of C2×C6.11D12 in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
12120000
100000
0012000
0001200
0000012
000011
,
360000
7100000
0091000
005400
000088
000050
,
360000
3100000
0091000
0010400
000050
000088

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,9,5,0,0,0,0,10,4,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,9,10,0,0,0,0,10,4,0,0,0,0,0,0,5,8,0,0,0,0,0,8] >;

C2×C6.11D12 in GAP, Magma, Sage, TeX

C_2\times C_6._{11}D_{12}
% in TeX

G:=Group("C2xC6.11D12");
// GroupNames label

G:=SmallGroup(288,784);
// by ID

G=gap.SmallGroup(288,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^12=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

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×
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