metabelian, supersoluble, monomial
Aliases: C122⋊2C2, C62.220C23, (C4×C12)⋊5S3, C42⋊3(C3⋊S3), (C2×C12).358D6, C3⋊3(C42⋊3S3), C6.96(C4○D12), C6.Dic6⋊1C2, (C6×C12).288C22, C32⋊9(C42⋊2C2), C6.11D12.1C2, C2.8(C12.59D6), (C3×C6).112(C4○D4), (C2×C6).237(C22×S3), C22.38(C22×C3⋊S3), (C22×C3⋊S3).39C22, (C2×C3⋊Dic3).77C22, (C2×C4).66(C2×C3⋊S3), SmallGroup(288,733)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C122⋊2C2
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b6, cbc=a6b-1 >
Subgroups: 684 in 180 conjugacy classes, 65 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C4×C12, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C42⋊3S3, C6.Dic6, C6.11D12, C122, C122⋊2C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C42⋊2C2, C2×C3⋊S3, C4○D12, C22×C3⋊S3, C42⋊3S3, C12.59D6, C122⋊2C2
►On 144 points
Generators in S144
(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)
(1 108 52 117 32 70 44 14 135 121 78 85)(2 97 53 118 33 71 45 15 136 122 79 86)(3 98 54 119 34 72 46 16 137 123 80 87)(4 99 55 120 35 61 47 17 138 124 81 88)(5 100 56 109 36 62 48 18 139 125 82 89)(6 101 57 110 25 63 37 19 140 126 83 90)(7 102 58 111 26 64 38 20 141 127 84 91)(8 103 59 112 27 65 39 21 142 128 73 92)(9 104 60 113 28 66 40 22 143 129 74 93)(10 105 49 114 29 67 41 23 144 130 75 94)(11 106 50 115 30 68 42 24 133 131 76 95)(12 107 51 116 31 69 43 13 134 132 77 96)
(2 37)(3 11)(4 47)(5 9)(6 45)(8 43)(10 41)(12 39)(13 86)(14 64)(15 96)(16 62)(17 94)(18 72)(19 92)(20 70)(21 90)(22 68)(23 88)(24 66)(25 53)(26 141)(27 51)(28 139)(29 49)(30 137)(31 59)(32 135)(33 57)(34 133)(35 55)(36 143)(40 48)(42 46)(50 80)(52 78)(54 76)(56 74)(58 84)(60 82)(61 105)(63 103)(65 101)(67 99)(69 97)(71 107)(73 134)(75 144)(77 142)(79 140)(81 138)(83 136)(85 102)(87 100)(89 98)(91 108)(93 106)(95 104)(109 123)(110 112)(111 121)(113 131)(114 120)(115 129)(116 118)(117 127)(119 125)(122 132)(124 130)(126 128)
G:=sub<Sym(144)| (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), (1,108,52,117,32,70,44,14,135,121,78,85)(2,97,53,118,33,71,45,15,136,122,79,86)(3,98,54,119,34,72,46,16,137,123,80,87)(4,99,55,120,35,61,47,17,138,124,81,88)(5,100,56,109,36,62,48,18,139,125,82,89)(6,101,57,110,25,63,37,19,140,126,83,90)(7,102,58,111,26,64,38,20,141,127,84,91)(8,103,59,112,27,65,39,21,142,128,73,92)(9,104,60,113,28,66,40,22,143,129,74,93)(10,105,49,114,29,67,41,23,144,130,75,94)(11,106,50,115,30,68,42,24,133,131,76,95)(12,107,51,116,31,69,43,13,134,132,77,96), (2,37)(3,11)(4,47)(5,9)(6,45)(8,43)(10,41)(12,39)(13,86)(14,64)(15,96)(16,62)(17,94)(18,72)(19,92)(20,70)(21,90)(22,68)(23,88)(24,66)(25,53)(26,141)(27,51)(28,139)(29,49)(30,137)(31,59)(32,135)(33,57)(34,133)(35,55)(36,143)(40,48)(42,46)(50,80)(52,78)(54,76)(56,74)(58,84)(60,82)(61,105)(63,103)(65,101)(67,99)(69,97)(71,107)(73,134)(75,144)(77,142)(79,140)(81,138)(83,136)(85,102)(87,100)(89,98)(91,108)(93,106)(95,104)(109,123)(110,112)(111,121)(113,131)(114,120)(115,129)(116,118)(117,127)(119,125)(122,132)(124,130)(126,128)>;
G:=Group( (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), (1,108,52,117,32,70,44,14,135,121,78,85)(2,97,53,118,33,71,45,15,136,122,79,86)(3,98,54,119,34,72,46,16,137,123,80,87)(4,99,55,120,35,61,47,17,138,124,81,88)(5,100,56,109,36,62,48,18,139,125,82,89)(6,101,57,110,25,63,37,19,140,126,83,90)(7,102,58,111,26,64,38,20,141,127,84,91)(8,103,59,112,27,65,39,21,142,128,73,92)(9,104,60,113,28,66,40,22,143,129,74,93)(10,105,49,114,29,67,41,23,144,130,75,94)(11,106,50,115,30,68,42,24,133,131,76,95)(12,107,51,116,31,69,43,13,134,132,77,96), (2,37)(3,11)(4,47)(5,9)(6,45)(8,43)(10,41)(12,39)(13,86)(14,64)(15,96)(16,62)(17,94)(18,72)(19,92)(20,70)(21,90)(22,68)(23,88)(24,66)(25,53)(26,141)(27,51)(28,139)(29,49)(30,137)(31,59)(32,135)(33,57)(34,133)(35,55)(36,143)(40,48)(42,46)(50,80)(52,78)(54,76)(56,74)(58,84)(60,82)(61,105)(63,103)(65,101)(67,99)(69,97)(71,107)(73,134)(75,144)(77,142)(79,140)(81,138)(83,136)(85,102)(87,100)(89,98)(91,108)(93,106)(95,104)(109,123)(110,112)(111,121)(113,131)(114,120)(115,129)(116,118)(117,127)(119,125)(122,132)(124,130)(126,128) );
G=PermutationGroup([[(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)], [(1,108,52,117,32,70,44,14,135,121,78,85),(2,97,53,118,33,71,45,15,136,122,79,86),(3,98,54,119,34,72,46,16,137,123,80,87),(4,99,55,120,35,61,47,17,138,124,81,88),(5,100,56,109,36,62,48,18,139,125,82,89),(6,101,57,110,25,63,37,19,140,126,83,90),(7,102,58,111,26,64,38,20,141,127,84,91),(8,103,59,112,27,65,39,21,142,128,73,92),(9,104,60,113,28,66,40,22,143,129,74,93),(10,105,49,114,29,67,41,23,144,130,75,94),(11,106,50,115,30,68,42,24,133,131,76,95),(12,107,51,116,31,69,43,13,134,132,77,96)], [(2,37),(3,11),(4,47),(5,9),(6,45),(8,43),(10,41),(12,39),(13,86),(14,64),(15,96),(16,62),(17,94),(18,72),(19,92),(20,70),(21,90),(22,68),(23,88),(24,66),(25,53),(26,141),(27,51),(28,139),(29,49),(30,137),(31,59),(32,135),(33,57),(34,133),(35,55),(36,143),(40,48),(42,46),(50,80),(52,78),(54,76),(56,74),(58,84),(60,82),(61,105),(63,103),(65,101),(67,99),(69,97),(71,107),(73,134),(75,144),(77,142),(79,140),(81,138),(83,136),(85,102),(87,100),(89,98),(91,108),(93,106),(95,104),(109,123),(110,112),(111,121),(113,131),(114,120),(115,129),(116,118),(117,127),(119,125),(122,132),(124,130),(126,128)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D6 | C4○D4 | C4○D12 |
| kernel | C122⋊2C2 | C6.Dic6 | C6.11D12 | C122 | C4×C12 | C2×C12 | C3×C6 | C6 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 6 | 48 |
Matrix representation of C122⋊2C2 ►in GL4(𝔽13) generated by
| 10 | 10 | 0 | 0 |
| 3 | 7 | 0 | 0 |
| 0 | 0 | 11 | 4 |
| 0 | 0 | 9 | 2 |
| 5 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 10 | 6 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [10,3,0,0,10,7,0,0,0,0,11,9,0,0,4,2],[5,5,0,0,8,0,0,0,0,0,3,10,0,0,3,6],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,1,1] >;
C122⋊2C2 in GAP, Magma, Sage, TeX
C_{12}^2\rtimes_2C_2 % in TeX
G:=Group("C12^2:2C2"); // GroupNames label
G:=SmallGroup(288,733);
// by ID
G=gap.SmallGroup(288,733);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5*b^6,c*b*c=a^6*b^-1>;
// generators/relations