metabelian, supersoluble, monomial
Aliases: C122⋊6C2, C12.47D12, C62.219C23, (C4×C12)⋊7S3, C42⋊7(C3⋊S3), C6.51(C2×D12), (C3×C12).122D4, (C2×C12).357D6, C4.5(C12⋊S3), C3⋊2(C42⋊7S3), C6.95(C4○D12), C6.11D12⋊1C2, (C6×C12).287C22, C32⋊12(C4.4D4), C2.7(C12.59D6), C2.6(C2×C12⋊S3), (C3×C6).191(C2×D4), (C2×C32⋊4Q8)⋊3C2, (C2×C12⋊S3).4C2, (C3×C6).111(C4○D4), (C2×C6).236(C22×S3), C22.37(C22×C3⋊S3), (C22×C3⋊S3).38C22, (C2×C3⋊Dic3).76C22, (C2×C4).65(C2×C3⋊S3), SmallGroup(288,732)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C122⋊6C2
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b6, cbc=a6b5 >
Subgroups: 1036 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C4.4D4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C42⋊7S3, C6.11D12, C122, C2×C32⋊4Q8, C2×C12⋊S3, C122⋊6C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, D12, C22×S3, C4.4D4, C2×C3⋊S3, C2×D12, C4○D12, C12⋊S3, C22×C3⋊S3, C42⋊7S3, C2×C12⋊S3, C12.59D6, C122⋊6C2
►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)
(1 119 34 72 76 46 49 133 97 131 92 20)(2 120 35 61 77 47 50 134 98 132 93 21)(3 109 36 62 78 48 51 135 99 121 94 22)(4 110 25 63 79 37 52 136 100 122 95 23)(5 111 26 64 80 38 53 137 101 123 96 24)(6 112 27 65 81 39 54 138 102 124 85 13)(7 113 28 66 82 40 55 139 103 125 86 14)(8 114 29 67 83 41 56 140 104 126 87 15)(9 115 30 68 84 42 57 141 105 127 88 16)(10 116 31 69 73 43 58 142 106 128 89 17)(11 117 32 70 74 44 59 143 107 129 90 18)(12 118 33 71 75 45 60 144 108 130 91 19)
(2 54)(3 11)(4 52)(5 9)(6 50)(8 60)(10 58)(12 56)(13 114)(14 133)(15 112)(16 143)(17 110)(18 141)(19 120)(20 139)(21 118)(22 137)(23 116)(24 135)(25 79)(26 88)(27 77)(28 86)(29 75)(30 96)(31 73)(32 94)(33 83)(34 92)(35 81)(36 90)(37 142)(38 109)(39 140)(40 119)(41 138)(42 117)(43 136)(44 115)(45 134)(46 113)(47 144)(48 111)(51 59)(53 57)(61 130)(62 64)(63 128)(65 126)(66 72)(67 124)(68 70)(69 122)(71 132)(74 99)(76 97)(78 107)(80 105)(82 103)(84 101)(85 98)(87 108)(89 106)(91 104)(93 102)(95 100)(121 123)(125 131)(127 129)
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), (1,119,34,72,76,46,49,133,97,131,92,20)(2,120,35,61,77,47,50,134,98,132,93,21)(3,109,36,62,78,48,51,135,99,121,94,22)(4,110,25,63,79,37,52,136,100,122,95,23)(5,111,26,64,80,38,53,137,101,123,96,24)(6,112,27,65,81,39,54,138,102,124,85,13)(7,113,28,66,82,40,55,139,103,125,86,14)(8,114,29,67,83,41,56,140,104,126,87,15)(9,115,30,68,84,42,57,141,105,127,88,16)(10,116,31,69,73,43,58,142,106,128,89,17)(11,117,32,70,74,44,59,143,107,129,90,18)(12,118,33,71,75,45,60,144,108,130,91,19), (2,54)(3,11)(4,52)(5,9)(6,50)(8,60)(10,58)(12,56)(13,114)(14,133)(15,112)(16,143)(17,110)(18,141)(19,120)(20,139)(21,118)(22,137)(23,116)(24,135)(25,79)(26,88)(27,77)(28,86)(29,75)(30,96)(31,73)(32,94)(33,83)(34,92)(35,81)(36,90)(37,142)(38,109)(39,140)(40,119)(41,138)(42,117)(43,136)(44,115)(45,134)(46,113)(47,144)(48,111)(51,59)(53,57)(61,130)(62,64)(63,128)(65,126)(66,72)(67,124)(68,70)(69,122)(71,132)(74,99)(76,97)(78,107)(80,105)(82,103)(84,101)(85,98)(87,108)(89,106)(91,104)(93,102)(95,100)(121,123)(125,131)(127,129)>;
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), (1,119,34,72,76,46,49,133,97,131,92,20)(2,120,35,61,77,47,50,134,98,132,93,21)(3,109,36,62,78,48,51,135,99,121,94,22)(4,110,25,63,79,37,52,136,100,122,95,23)(5,111,26,64,80,38,53,137,101,123,96,24)(6,112,27,65,81,39,54,138,102,124,85,13)(7,113,28,66,82,40,55,139,103,125,86,14)(8,114,29,67,83,41,56,140,104,126,87,15)(9,115,30,68,84,42,57,141,105,127,88,16)(10,116,31,69,73,43,58,142,106,128,89,17)(11,117,32,70,74,44,59,143,107,129,90,18)(12,118,33,71,75,45,60,144,108,130,91,19), (2,54)(3,11)(4,52)(5,9)(6,50)(8,60)(10,58)(12,56)(13,114)(14,133)(15,112)(16,143)(17,110)(18,141)(19,120)(20,139)(21,118)(22,137)(23,116)(24,135)(25,79)(26,88)(27,77)(28,86)(29,75)(30,96)(31,73)(32,94)(33,83)(34,92)(35,81)(36,90)(37,142)(38,109)(39,140)(40,119)(41,138)(42,117)(43,136)(44,115)(45,134)(46,113)(47,144)(48,111)(51,59)(53,57)(61,130)(62,64)(63,128)(65,126)(66,72)(67,124)(68,70)(69,122)(71,132)(74,99)(76,97)(78,107)(80,105)(82,103)(84,101)(85,98)(87,108)(89,106)(91,104)(93,102)(95,100)(121,123)(125,131)(127,129) );
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)], [(1,119,34,72,76,46,49,133,97,131,92,20),(2,120,35,61,77,47,50,134,98,132,93,21),(3,109,36,62,78,48,51,135,99,121,94,22),(4,110,25,63,79,37,52,136,100,122,95,23),(5,111,26,64,80,38,53,137,101,123,96,24),(6,112,27,65,81,39,54,138,102,124,85,13),(7,113,28,66,82,40,55,139,103,125,86,14),(8,114,29,67,83,41,56,140,104,126,87,15),(9,115,30,68,84,42,57,141,105,127,88,16),(10,116,31,69,73,43,58,142,106,128,89,17),(11,117,32,70,74,44,59,143,107,129,90,18),(12,118,33,71,75,45,60,144,108,130,91,19)], [(2,54),(3,11),(4,52),(5,9),(6,50),(8,60),(10,58),(12,56),(13,114),(14,133),(15,112),(16,143),(17,110),(18,141),(19,120),(20,139),(21,118),(22,137),(23,116),(24,135),(25,79),(26,88),(27,77),(28,86),(29,75),(30,96),(31,73),(32,94),(33,83),(34,92),(35,81),(36,90),(37,142),(38,109),(39,140),(40,119),(41,138),(42,117),(43,136),(44,115),(45,134),(46,113),(47,144),(48,111),(51,59),(53,57),(61,130),(62,64),(63,128),(65,126),(66,72),(67,124),(68,70),(69,122),(71,132),(74,99),(76,97),(78,107),(80,105),(82,103),(84,101),(85,98),(87,108),(89,106),(91,104),(93,102),(95,100),(121,123),(125,131),(127,129)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | D12 | C4○D12 |
| kernel | C122⋊6C2 | C6.11D12 | C122 | C2×C32⋊4Q8 | C2×C12⋊S3 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 12 | 4 | 16 | 32 |
Matrix representation of C122⋊6C2 ►in GL4(𝔽13) generated by
| 0 | 5 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 11 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [0,8,0,0,5,8,0,0,0,0,9,11,0,0,2,11],[11,4,0,0,9,2,0,0,0,0,5,5,0,0,8,0],[1,12,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C122⋊6C2 in GAP, Magma, Sage, TeX
C_{12}^2\rtimes_6C_2 % in TeX
G:=Group("C12^2:6C2"); // GroupNames label
G:=SmallGroup(288,732);
// by ID
G=gap.SmallGroup(288,732);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,100,2693,9414]);
// 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^5>;
// generators/relations