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 [×3], C2, C3 [×4], C4 [×6], C22, C22 [×3], S3 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×3], C23, C32, Dic3 [×12], C12 [×12], D6 [×12], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3⋊S3, C3×C6 [×3], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C42⋊2C2, C3⋊Dic3 [×3], C3×C12 [×3], C2×C3⋊S3 [×3], C62, Dic3⋊C4 [×12], D6⋊C4 [×12], C4×C12 [×4], C2×C3⋊Dic3 [×3], C6×C12 [×3], C22×C3⋊S3, C42⋊3S3 [×4], C6.Dic6 [×3], C6.11D12 [×3], C122, C122⋊2C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4 [×3], C3⋊S3, C22×S3 [×4], C42⋊2C2, C2×C3⋊S3 [×3], C4○D12 [×12], C22×C3⋊S3, C42⋊3S3 [×4], C12.59D6 [×3], 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 14 79 42 105 62 91 121 26 142 59 113)(2 15 80 43 106 63 92 122 27 143 60 114)(3 16 81 44 107 64 93 123 28 144 49 115)(4 17 82 45 108 65 94 124 29 133 50 116)(5 18 83 46 97 66 95 125 30 134 51 117)(6 19 84 47 98 67 96 126 31 135 52 118)(7 20 73 48 99 68 85 127 32 136 53 119)(8 21 74 37 100 69 86 128 33 137 54 120)(9 22 75 38 101 70 87 129 34 138 55 109)(10 23 76 39 102 71 88 130 35 139 56 110)(11 24 77 40 103 72 89 131 36 140 57 111)(12 13 78 41 104 61 90 132 25 141 58 112)
(2 96)(3 11)(4 94)(5 9)(6 92)(8 90)(10 88)(12 86)(13 63)(14 119)(15 61)(16 117)(17 71)(18 115)(19 69)(20 113)(21 67)(22 111)(23 65)(24 109)(25 54)(26 105)(27 52)(28 103)(29 50)(30 101)(31 60)(32 99)(33 58)(34 97)(35 56)(36 107)(37 47)(38 140)(39 45)(40 138)(41 43)(42 136)(44 134)(46 144)(48 142)(49 77)(51 75)(53 73)(55 83)(57 81)(59 79)(62 127)(64 125)(66 123)(68 121)(70 131)(72 129)(74 104)(76 102)(78 100)(80 98)(82 108)(84 106)(87 95)(89 93)(110 124)(112 122)(114 132)(116 130)(118 128)(120 126)(133 139)(135 137)(141 143)
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), (1,14,79,42,105,62,91,121,26,142,59,113)(2,15,80,43,106,63,92,122,27,143,60,114)(3,16,81,44,107,64,93,123,28,144,49,115)(4,17,82,45,108,65,94,124,29,133,50,116)(5,18,83,46,97,66,95,125,30,134,51,117)(6,19,84,47,98,67,96,126,31,135,52,118)(7,20,73,48,99,68,85,127,32,136,53,119)(8,21,74,37,100,69,86,128,33,137,54,120)(9,22,75,38,101,70,87,129,34,138,55,109)(10,23,76,39,102,71,88,130,35,139,56,110)(11,24,77,40,103,72,89,131,36,140,57,111)(12,13,78,41,104,61,90,132,25,141,58,112), (2,96)(3,11)(4,94)(5,9)(6,92)(8,90)(10,88)(12,86)(13,63)(14,119)(15,61)(16,117)(17,71)(18,115)(19,69)(20,113)(21,67)(22,111)(23,65)(24,109)(25,54)(26,105)(27,52)(28,103)(29,50)(30,101)(31,60)(32,99)(33,58)(34,97)(35,56)(36,107)(37,47)(38,140)(39,45)(40,138)(41,43)(42,136)(44,134)(46,144)(48,142)(49,77)(51,75)(53,73)(55,83)(57,81)(59,79)(62,127)(64,125)(66,123)(68,121)(70,131)(72,129)(74,104)(76,102)(78,100)(80,98)(82,108)(84,106)(87,95)(89,93)(110,124)(112,122)(114,132)(116,130)(118,128)(120,126)(133,139)(135,137)(141,143)>;
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), (1,14,79,42,105,62,91,121,26,142,59,113)(2,15,80,43,106,63,92,122,27,143,60,114)(3,16,81,44,107,64,93,123,28,144,49,115)(4,17,82,45,108,65,94,124,29,133,50,116)(5,18,83,46,97,66,95,125,30,134,51,117)(6,19,84,47,98,67,96,126,31,135,52,118)(7,20,73,48,99,68,85,127,32,136,53,119)(8,21,74,37,100,69,86,128,33,137,54,120)(9,22,75,38,101,70,87,129,34,138,55,109)(10,23,76,39,102,71,88,130,35,139,56,110)(11,24,77,40,103,72,89,131,36,140,57,111)(12,13,78,41,104,61,90,132,25,141,58,112), (2,96)(3,11)(4,94)(5,9)(6,92)(8,90)(10,88)(12,86)(13,63)(14,119)(15,61)(16,117)(17,71)(18,115)(19,69)(20,113)(21,67)(22,111)(23,65)(24,109)(25,54)(26,105)(27,52)(28,103)(29,50)(30,101)(31,60)(32,99)(33,58)(34,97)(35,56)(36,107)(37,47)(38,140)(39,45)(40,138)(41,43)(42,136)(44,134)(46,144)(48,142)(49,77)(51,75)(53,73)(55,83)(57,81)(59,79)(62,127)(64,125)(66,123)(68,121)(70,131)(72,129)(74,104)(76,102)(78,100)(80,98)(82,108)(84,106)(87,95)(89,93)(110,124)(112,122)(114,132)(116,130)(118,128)(120,126)(133,139)(135,137)(141,143) );
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)], [(1,14,79,42,105,62,91,121,26,142,59,113),(2,15,80,43,106,63,92,122,27,143,60,114),(3,16,81,44,107,64,93,123,28,144,49,115),(4,17,82,45,108,65,94,124,29,133,50,116),(5,18,83,46,97,66,95,125,30,134,51,117),(6,19,84,47,98,67,96,126,31,135,52,118),(7,20,73,48,99,68,85,127,32,136,53,119),(8,21,74,37,100,69,86,128,33,137,54,120),(9,22,75,38,101,70,87,129,34,138,55,109),(10,23,76,39,102,71,88,130,35,139,56,110),(11,24,77,40,103,72,89,131,36,140,57,111),(12,13,78,41,104,61,90,132,25,141,58,112)], [(2,96),(3,11),(4,94),(5,9),(6,92),(8,90),(10,88),(12,86),(13,63),(14,119),(15,61),(16,117),(17,71),(18,115),(19,69),(20,113),(21,67),(22,111),(23,65),(24,109),(25,54),(26,105),(27,52),(28,103),(29,50),(30,101),(31,60),(32,99),(33,58),(34,97),(35,56),(36,107),(37,47),(38,140),(39,45),(40,138),(41,43),(42,136),(44,134),(46,144),(48,142),(49,77),(51,75),(53,73),(55,83),(57,81),(59,79),(62,127),(64,125),(66,123),(68,121),(70,131),(72,129),(74,104),(76,102),(78,100),(80,98),(82,108),(84,106),(87,95),(89,93),(110,124),(112,122),(114,132),(116,130),(118,128),(120,126),(133,139),(135,137),(141,143)])
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