metabelian, supersoluble, monomial
Aliases: C12⋊1Dic3, C6.10D12, C6.7Dic6, C62.23C22, (C3×C12)⋊3C4, C4⋊(C3⋊Dic3), (C3×C6).7Q8, C32⋊9(C4⋊C4), (C6×C12).4C2, (C2×C12).8S3, (C3×C6).25D4, (C2×C6).32D6, C3⋊2(C4⋊Dic3), C6.15(C2×Dic3), C2.1(C12⋊S3), C2.2(C32⋊4Q8), (C2×C4).3(C3⋊S3), (C3×C6).33(C2×C4), C22.5(C2×C3⋊S3), C2.4(C2×C3⋊Dic3), (C2×C3⋊Dic3).5C2, SmallGroup(144,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊Dic3
G = < a,b,c | a12=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 202 in 78 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C2×C3⋊Dic3, C6×C12, C12⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, D12, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊Dic3
►Regular action 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 77 27 67 14 54)(2 78 28 68 15 55)(3 79 29 69 16 56)(4 80 30 70 17 57)(5 81 31 71 18 58)(6 82 32 72 19 59)(7 83 33 61 20 60)(8 84 34 62 21 49)(9 73 35 63 22 50)(10 74 36 64 23 51)(11 75 25 65 24 52)(12 76 26 66 13 53)(37 109 132 108 144 92)(38 110 121 97 133 93)(39 111 122 98 134 94)(40 112 123 99 135 95)(41 113 124 100 136 96)(42 114 125 101 137 85)(43 115 126 102 138 86)(44 116 127 103 139 87)(45 117 128 104 140 88)(46 118 129 105 141 89)(47 119 130 106 142 90)(48 120 131 107 143 91)
(1 122 67 94)(2 121 68 93)(3 132 69 92)(4 131 70 91)(5 130 71 90)(6 129 72 89)(7 128 61 88)(8 127 62 87)(9 126 63 86)(10 125 64 85)(11 124 65 96)(12 123 66 95)(13 135 76 112)(14 134 77 111)(15 133 78 110)(16 144 79 109)(17 143 80 120)(18 142 81 119)(19 141 82 118)(20 140 83 117)(21 139 84 116)(22 138 73 115)(23 137 74 114)(24 136 75 113)(25 41 52 100)(26 40 53 99)(27 39 54 98)(28 38 55 97)(29 37 56 108)(30 48 57 107)(31 47 58 106)(32 46 59 105)(33 45 60 104)(34 44 49 103)(35 43 50 102)(36 42 51 101)
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), (1,77,27,67,14,54)(2,78,28,68,15,55)(3,79,29,69,16,56)(4,80,30,70,17,57)(5,81,31,71,18,58)(6,82,32,72,19,59)(7,83,33,61,20,60)(8,84,34,62,21,49)(9,73,35,63,22,50)(10,74,36,64,23,51)(11,75,25,65,24,52)(12,76,26,66,13,53)(37,109,132,108,144,92)(38,110,121,97,133,93)(39,111,122,98,134,94)(40,112,123,99,135,95)(41,113,124,100,136,96)(42,114,125,101,137,85)(43,115,126,102,138,86)(44,116,127,103,139,87)(45,117,128,104,140,88)(46,118,129,105,141,89)(47,119,130,106,142,90)(48,120,131,107,143,91), (1,122,67,94)(2,121,68,93)(3,132,69,92)(4,131,70,91)(5,130,71,90)(6,129,72,89)(7,128,61,88)(8,127,62,87)(9,126,63,86)(10,125,64,85)(11,124,65,96)(12,123,66,95)(13,135,76,112)(14,134,77,111)(15,133,78,110)(16,144,79,109)(17,143,80,120)(18,142,81,119)(19,141,82,118)(20,140,83,117)(21,139,84,116)(22,138,73,115)(23,137,74,114)(24,136,75,113)(25,41,52,100)(26,40,53,99)(27,39,54,98)(28,38,55,97)(29,37,56,108)(30,48,57,107)(31,47,58,106)(32,46,59,105)(33,45,60,104)(34,44,49,103)(35,43,50,102)(36,42,51,101)>;
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), (1,77,27,67,14,54)(2,78,28,68,15,55)(3,79,29,69,16,56)(4,80,30,70,17,57)(5,81,31,71,18,58)(6,82,32,72,19,59)(7,83,33,61,20,60)(8,84,34,62,21,49)(9,73,35,63,22,50)(10,74,36,64,23,51)(11,75,25,65,24,52)(12,76,26,66,13,53)(37,109,132,108,144,92)(38,110,121,97,133,93)(39,111,122,98,134,94)(40,112,123,99,135,95)(41,113,124,100,136,96)(42,114,125,101,137,85)(43,115,126,102,138,86)(44,116,127,103,139,87)(45,117,128,104,140,88)(46,118,129,105,141,89)(47,119,130,106,142,90)(48,120,131,107,143,91), (1,122,67,94)(2,121,68,93)(3,132,69,92)(4,131,70,91)(5,130,71,90)(6,129,72,89)(7,128,61,88)(8,127,62,87)(9,126,63,86)(10,125,64,85)(11,124,65,96)(12,123,66,95)(13,135,76,112)(14,134,77,111)(15,133,78,110)(16,144,79,109)(17,143,80,120)(18,142,81,119)(19,141,82,118)(20,140,83,117)(21,139,84,116)(22,138,73,115)(23,137,74,114)(24,136,75,113)(25,41,52,100)(26,40,53,99)(27,39,54,98)(28,38,55,97)(29,37,56,108)(30,48,57,107)(31,47,58,106)(32,46,59,105)(33,45,60,104)(34,44,49,103)(35,43,50,102)(36,42,51,101) );
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)], [(1,77,27,67,14,54),(2,78,28,68,15,55),(3,79,29,69,16,56),(4,80,30,70,17,57),(5,81,31,71,18,58),(6,82,32,72,19,59),(7,83,33,61,20,60),(8,84,34,62,21,49),(9,73,35,63,22,50),(10,74,36,64,23,51),(11,75,25,65,24,52),(12,76,26,66,13,53),(37,109,132,108,144,92),(38,110,121,97,133,93),(39,111,122,98,134,94),(40,112,123,99,135,95),(41,113,124,100,136,96),(42,114,125,101,137,85),(43,115,126,102,138,86),(44,116,127,103,139,87),(45,117,128,104,140,88),(46,118,129,105,141,89),(47,119,130,106,142,90),(48,120,131,107,143,91)], [(1,122,67,94),(2,121,68,93),(3,132,69,92),(4,131,70,91),(5,130,71,90),(6,129,72,89),(7,128,61,88),(8,127,62,87),(9,126,63,86),(10,125,64,85),(11,124,65,96),(12,123,66,95),(13,135,76,112),(14,134,77,111),(15,133,78,110),(16,144,79,109),(17,143,80,120),(18,142,81,119),(19,141,82,118),(20,140,83,117),(21,139,84,116),(22,138,73,115),(23,137,74,114),(24,136,75,113),(25,41,52,100),(26,40,53,99),(27,39,54,98),(28,38,55,97),(29,37,56,108),(30,48,57,107),(31,47,58,106),(32,46,59,105),(33,45,60,104),(34,44,49,103),(35,43,50,102),(36,42,51,101)]])
C12⋊Dic3 is a maximal subgroup of
C6.16D24 C6.Dic12 C12.Dic6 C6.18D24 C12.9Dic6 C12.10Dic6 C6.4Dic12 C24⋊2Dic3 C24⋊1Dic3 C62.84D4 C62.116D4 C62.117D4 C62.11C23 Dic3×Dic6 Dic3.Dic6 D6⋊6Dic6 C62.31C23 C62.39C23 S3×C4⋊Dic3 D6.9D12 Dic3×D12 D6⋊2Dic6 D6⋊2D12 C12⋊3Dic6 C4×C32⋊4Q8 C12⋊6Dic6 C12.25Dic6 C4×C12⋊S3 C62⋊6Q8 C62.223C23 C62.227C23 C62.69D4 C12⋊2Dic6 C62.233C23 C62.234C23 C4⋊C4×C3⋊S3 C62.236C23 C12.31D12 C62.242C23 C62⋊10Q8 C62.247C23 C62⋊19D4 D4×C3⋊Dic3 C62.256C23 Q8×C3⋊Dic3 C62.261C23 C62.20D6 C36⋊Dic3 C62.80D6 C62.147D6
C12⋊Dic3 is a maximal quotient of
C12.57D12 C24⋊2Dic3 C24⋊1Dic3 C12.59D12 C62.15Q8 C36⋊Dic3 C62.30D6 C62.80D6 C62.147D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | D12 |
| kernel | C12⋊Dic3 | C2×C3⋊Dic3 | C6×C12 | C3×C12 | C2×C12 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 8 | 4 | 8 | 8 |
Matrix representation of C12⋊Dic3 ►in GL6(𝔽13)
| 10 | 10 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 3 | 3 |
G:=sub<GL(6,GF(13))| [10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8,0,0,0,0,0,0,10,3,0,0,0,0,6,3] >;
C12⋊Dic3 in GAP, Magma, Sage, TeX
C_{12}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C12:Dic3"); // GroupNames label
G:=SmallGroup(144,94);
// by ID
G=gap.SmallGroup(144,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations