direct product, metabelian, supersoluble, monomial
Aliases: C2×C6.11D12, C62.95D4, C62.248C23, C6⋊2(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(C32⋊7D4), (C2×C6×C12)⋊3C2, C3⋊3(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), C32⋊12(C2×C22⋊C4), C2.2(C2×C32⋊7D4), (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
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, C32⋊7D4, C22×C3⋊S3, C2×D6⋊C4, C6.11D12, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C32⋊7D4, 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 | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C2×C6.11D12 | C6.11D12 | C22×C3⋊Dic3 | C2×C6×C12 | C23×C3⋊S3 | C22×C3⋊S3 | C22×C12 | C62 | C2×C12 | C22×C6 | C2×C6 | C2×C6 | C2×C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 4 | 8 | 4 | 16 | 16 | 16 |
Matrix representation of C2×C6.11D12 ►in GL6(𝔽13)
| 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 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 5 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
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