metabelian, supersoluble, monomial
Aliases: C12.31D12, C62.241C23, C6.47(S3×Q8), (C3×C12).94D4, (C2×C12).35D6, C6.55(C2×D12), C3⋊3(C4.D12), C12⋊Dic3⋊9C2, (C6×C12).18C22, C4.13(C12⋊S3), C32⋊21(C22⋊Q8), C6.11D12.3C2, C6.102(D4⋊2S3), C2.13(C12.D6), (C3×C4⋊C4)⋊8S3, C4⋊C4⋊5(C3⋊S3), (C2×C3⋊S3)⋊8Q8, C2.7(Q8×C3⋊S3), (C3×C6).74(C2×Q8), (C32×C4⋊C4)⋊17C2, (C3×C6).195(C2×D4), C2.10(C2×C12⋊S3), (C2×C32⋊4Q8)⋊13C2, (C3×C6).149(C4○D4), (C2×C6).258(C22×S3), C22.52(C22×C3⋊S3), (C22×C3⋊S3).89C22, (C2×C3⋊Dic3).87C22, (C2×C4×C3⋊S3).6C2, (C2×C4).14(C2×C3⋊S3), SmallGroup(288,754)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.31D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 860 in 222 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊Q8, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C4.D12, C12⋊Dic3, C6.11D12, C32×C4⋊C4, C2×C32⋊4Q8, C2×C4×C3⋊S3, C12.31D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, D12, C22×S3, C22⋊Q8, C2×C3⋊S3, C2×D12, D4⋊2S3, S3×Q8, C12⋊S3, C22×C3⋊S3, C4.D12, C2×C12⋊S3, C12.D6, Q8×C3⋊S3, C12.31D12
►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 86 82 14 123 108 37 141 54 35 62 120)(2 93 83 21 124 103 38 136 55 30 63 115)(3 88 84 16 125 98 39 143 56 25 64 110)(4 95 73 23 126 105 40 138 57 32 65 117)(5 90 74 18 127 100 41 133 58 27 66 112)(6 85 75 13 128 107 42 140 59 34 67 119)(7 92 76 20 129 102 43 135 60 29 68 114)(8 87 77 15 130 97 44 142 49 36 69 109)(9 94 78 22 131 104 45 137 50 31 70 116)(10 89 79 17 132 99 46 144 51 26 71 111)(11 96 80 24 121 106 47 139 52 33 72 118)(12 91 81 19 122 101 48 134 53 28 61 113)
(1 114 7 120)(2 113 8 119)(3 112 9 118)(4 111 10 117)(5 110 11 116)(6 109 12 115)(13 55 19 49)(14 54 20 60)(15 53 21 59)(16 52 22 58)(17 51 23 57)(18 50 24 56)(25 80 31 74)(26 79 32 73)(27 78 33 84)(28 77 34 83)(29 76 35 82)(30 75 36 81)(37 102 43 108)(38 101 44 107)(39 100 45 106)(40 99 46 105)(41 98 47 104)(42 97 48 103)(61 93 67 87)(62 92 68 86)(63 91 69 85)(64 90 70 96)(65 89 71 95)(66 88 72 94)(121 137 127 143)(122 136 128 142)(123 135 129 141)(124 134 130 140)(125 133 131 139)(126 144 132 138)
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), (1,86,82,14,123,108,37,141,54,35,62,120)(2,93,83,21,124,103,38,136,55,30,63,115)(3,88,84,16,125,98,39,143,56,25,64,110)(4,95,73,23,126,105,40,138,57,32,65,117)(5,90,74,18,127,100,41,133,58,27,66,112)(6,85,75,13,128,107,42,140,59,34,67,119)(7,92,76,20,129,102,43,135,60,29,68,114)(8,87,77,15,130,97,44,142,49,36,69,109)(9,94,78,22,131,104,45,137,50,31,70,116)(10,89,79,17,132,99,46,144,51,26,71,111)(11,96,80,24,121,106,47,139,52,33,72,118)(12,91,81,19,122,101,48,134,53,28,61,113), (1,114,7,120)(2,113,8,119)(3,112,9,118)(4,111,10,117)(5,110,11,116)(6,109,12,115)(13,55,19,49)(14,54,20,60)(15,53,21,59)(16,52,22,58)(17,51,23,57)(18,50,24,56)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,102,43,108)(38,101,44,107)(39,100,45,106)(40,99,46,105)(41,98,47,104)(42,97,48,103)(61,93,67,87)(62,92,68,86)(63,91,69,85)(64,90,70,96)(65,89,71,95)(66,88,72,94)(121,137,127,143)(122,136,128,142)(123,135,129,141)(124,134,130,140)(125,133,131,139)(126,144,132,138)>;
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), (1,86,82,14,123,108,37,141,54,35,62,120)(2,93,83,21,124,103,38,136,55,30,63,115)(3,88,84,16,125,98,39,143,56,25,64,110)(4,95,73,23,126,105,40,138,57,32,65,117)(5,90,74,18,127,100,41,133,58,27,66,112)(6,85,75,13,128,107,42,140,59,34,67,119)(7,92,76,20,129,102,43,135,60,29,68,114)(8,87,77,15,130,97,44,142,49,36,69,109)(9,94,78,22,131,104,45,137,50,31,70,116)(10,89,79,17,132,99,46,144,51,26,71,111)(11,96,80,24,121,106,47,139,52,33,72,118)(12,91,81,19,122,101,48,134,53,28,61,113), (1,114,7,120)(2,113,8,119)(3,112,9,118)(4,111,10,117)(5,110,11,116)(6,109,12,115)(13,55,19,49)(14,54,20,60)(15,53,21,59)(16,52,22,58)(17,51,23,57)(18,50,24,56)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,102,43,108)(38,101,44,107)(39,100,45,106)(40,99,46,105)(41,98,47,104)(42,97,48,103)(61,93,67,87)(62,92,68,86)(63,91,69,85)(64,90,70,96)(65,89,71,95)(66,88,72,94)(121,137,127,143)(122,136,128,142)(123,135,129,141)(124,134,130,140)(125,133,131,139)(126,144,132,138) );
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)], [(1,86,82,14,123,108,37,141,54,35,62,120),(2,93,83,21,124,103,38,136,55,30,63,115),(3,88,84,16,125,98,39,143,56,25,64,110),(4,95,73,23,126,105,40,138,57,32,65,117),(5,90,74,18,127,100,41,133,58,27,66,112),(6,85,75,13,128,107,42,140,59,34,67,119),(7,92,76,20,129,102,43,135,60,29,68,114),(8,87,77,15,130,97,44,142,49,36,69,109),(9,94,78,22,131,104,45,137,50,31,70,116),(10,89,79,17,132,99,46,144,51,26,71,111),(11,96,80,24,121,106,47,139,52,33,72,118),(12,91,81,19,122,101,48,134,53,28,61,113)], [(1,114,7,120),(2,113,8,119),(3,112,9,118),(4,111,10,117),(5,110,11,116),(6,109,12,115),(13,55,19,49),(14,54,20,60),(15,53,21,59),(16,52,22,58),(17,51,23,57),(18,50,24,56),(25,80,31,74),(26,79,32,73),(27,78,33,84),(28,77,34,83),(29,76,35,82),(30,75,36,81),(37,102,43,108),(38,101,44,107),(39,100,45,106),(40,99,46,105),(41,98,47,104),(42,97,48,103),(61,93,67,87),(62,92,68,86),(63,91,69,85),(64,90,70,96),(65,89,71,95),(66,88,72,94),(121,137,127,143),(122,136,128,142),(123,135,129,141),(124,134,130,140),(125,133,131,139),(126,144,132,138)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 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 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | C4○D4 | D12 | D4⋊2S3 | S3×Q8 |
| kernel | C12.31D12 | C12⋊Dic3 | C6.11D12 | C32×C4⋊C4 | C2×C32⋊4Q8 | C2×C4×C3⋊S3 | C3×C4⋊C4 | C3×C12 | C2×C3⋊S3 | C2×C12 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C12.31D12 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C12.31D12 in GAP, Magma, Sage, TeX
C_{12}._{31}D_{12} % in TeX
G:=Group("C12.31D12"); // GroupNames label
G:=SmallGroup(288,754);
// by ID
G=gap.SmallGroup(288,754);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,142,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations