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 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×8], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C12⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊Dic3
►Regular action on 144 points
(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)
(1 126 46 64 29 135)(2 127 47 65 30 136)(3 128 48 66 31 137)(4 129 37 67 32 138)(5 130 38 68 33 139)(6 131 39 69 34 140)(7 132 40 70 35 141)(8 121 41 71 36 142)(9 122 42 72 25 143)(10 123 43 61 26 144)(11 124 44 62 27 133)(12 125 45 63 28 134)(13 95 51 81 104 117)(14 96 52 82 105 118)(15 85 53 83 106 119)(16 86 54 84 107 120)(17 87 55 73 108 109)(18 88 56 74 97 110)(19 89 57 75 98 111)(20 90 58 76 99 112)(21 91 59 77 100 113)(22 92 60 78 101 114)(23 93 49 79 102 115)(24 94 50 80 103 116)
(1 108 64 87)(2 107 65 86)(3 106 66 85)(4 105 67 96)(5 104 68 95)(6 103 69 94)(7 102 70 93)(8 101 71 92)(9 100 72 91)(10 99 61 90)(11 98 62 89)(12 97 63 88)(13 130 81 33)(14 129 82 32)(15 128 83 31)(16 127 84 30)(17 126 73 29)(18 125 74 28)(19 124 75 27)(20 123 76 26)(21 122 77 25)(22 121 78 36)(23 132 79 35)(24 131 80 34)(37 52 138 118)(38 51 139 117)(39 50 140 116)(40 49 141 115)(41 60 142 114)(42 59 143 113)(43 58 144 112)(44 57 133 111)(45 56 134 110)(46 55 135 109)(47 54 136 120)(48 53 137 119)
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), (1,126,46,64,29,135)(2,127,47,65,30,136)(3,128,48,66,31,137)(4,129,37,67,32,138)(5,130,38,68,33,139)(6,131,39,69,34,140)(7,132,40,70,35,141)(8,121,41,71,36,142)(9,122,42,72,25,143)(10,123,43,61,26,144)(11,124,44,62,27,133)(12,125,45,63,28,134)(13,95,51,81,104,117)(14,96,52,82,105,118)(15,85,53,83,106,119)(16,86,54,84,107,120)(17,87,55,73,108,109)(18,88,56,74,97,110)(19,89,57,75,98,111)(20,90,58,76,99,112)(21,91,59,77,100,113)(22,92,60,78,101,114)(23,93,49,79,102,115)(24,94,50,80,103,116), (1,108,64,87)(2,107,65,86)(3,106,66,85)(4,105,67,96)(5,104,68,95)(6,103,69,94)(7,102,70,93)(8,101,71,92)(9,100,72,91)(10,99,61,90)(11,98,62,89)(12,97,63,88)(13,130,81,33)(14,129,82,32)(15,128,83,31)(16,127,84,30)(17,126,73,29)(18,125,74,28)(19,124,75,27)(20,123,76,26)(21,122,77,25)(22,121,78,36)(23,132,79,35)(24,131,80,34)(37,52,138,118)(38,51,139,117)(39,50,140,116)(40,49,141,115)(41,60,142,114)(42,59,143,113)(43,58,144,112)(44,57,133,111)(45,56,134,110)(46,55,135,109)(47,54,136,120)(48,53,137,119)>;
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), (1,126,46,64,29,135)(2,127,47,65,30,136)(3,128,48,66,31,137)(4,129,37,67,32,138)(5,130,38,68,33,139)(6,131,39,69,34,140)(7,132,40,70,35,141)(8,121,41,71,36,142)(9,122,42,72,25,143)(10,123,43,61,26,144)(11,124,44,62,27,133)(12,125,45,63,28,134)(13,95,51,81,104,117)(14,96,52,82,105,118)(15,85,53,83,106,119)(16,86,54,84,107,120)(17,87,55,73,108,109)(18,88,56,74,97,110)(19,89,57,75,98,111)(20,90,58,76,99,112)(21,91,59,77,100,113)(22,92,60,78,101,114)(23,93,49,79,102,115)(24,94,50,80,103,116), (1,108,64,87)(2,107,65,86)(3,106,66,85)(4,105,67,96)(5,104,68,95)(6,103,69,94)(7,102,70,93)(8,101,71,92)(9,100,72,91)(10,99,61,90)(11,98,62,89)(12,97,63,88)(13,130,81,33)(14,129,82,32)(15,128,83,31)(16,127,84,30)(17,126,73,29)(18,125,74,28)(19,124,75,27)(20,123,76,26)(21,122,77,25)(22,121,78,36)(23,132,79,35)(24,131,80,34)(37,52,138,118)(38,51,139,117)(39,50,140,116)(40,49,141,115)(41,60,142,114)(42,59,143,113)(43,58,144,112)(44,57,133,111)(45,56,134,110)(46,55,135,109)(47,54,136,120)(48,53,137,119) );
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)], [(1,126,46,64,29,135),(2,127,47,65,30,136),(3,128,48,66,31,137),(4,129,37,67,32,138),(5,130,38,68,33,139),(6,131,39,69,34,140),(7,132,40,70,35,141),(8,121,41,71,36,142),(9,122,42,72,25,143),(10,123,43,61,26,144),(11,124,44,62,27,133),(12,125,45,63,28,134),(13,95,51,81,104,117),(14,96,52,82,105,118),(15,85,53,83,106,119),(16,86,54,84,107,120),(17,87,55,73,108,109),(18,88,56,74,97,110),(19,89,57,75,98,111),(20,90,58,76,99,112),(21,91,59,77,100,113),(22,92,60,78,101,114),(23,93,49,79,102,115),(24,94,50,80,103,116)], [(1,108,64,87),(2,107,65,86),(3,106,66,85),(4,105,67,96),(5,104,68,95),(6,103,69,94),(7,102,70,93),(8,101,71,92),(9,100,72,91),(10,99,61,90),(11,98,62,89),(12,97,63,88),(13,130,81,33),(14,129,82,32),(15,128,83,31),(16,127,84,30),(17,126,73,29),(18,125,74,28),(19,124,75,27),(20,123,76,26),(21,122,77,25),(22,121,78,36),(23,132,79,35),(24,131,80,34),(37,52,138,118),(38,51,139,117),(39,50,140,116),(40,49,141,115),(41,60,142,114),(42,59,143,113),(43,58,144,112),(44,57,133,111),(45,56,134,110),(46,55,135,109),(47,54,136,120),(48,53,137,119)])
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