metabelian, supersoluble, monomial
Aliases: C6.6Dic6, C62.22C22, (C3×C6).6Q8, C6.14(C4×S3), C32⋊8(C4⋊C4), (C2×C12).4S3, (C6×C12).2C2, C3⋊Dic3⋊3C4, (C3×C6).33D4, (C2×C6).31D6, C3⋊3(Dic3⋊C4), C6.20(C3⋊D4), C2.1(C32⋊7D4), C2.1(C32⋊4Q8), C2.4(C4×C3⋊S3), (C2×C4).1(C3⋊S3), (C3×C6).25(C2×C4), C22.4(C2×C3⋊S3), (C2×C3⋊Dic3).4C2, SmallGroup(144,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Dic6
G = < a,b,c | a6=b12=1, c2=a3b6, ab=ba, cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 202 in 78 conjugacy classes, 41 normal (15 characteristic)
C1, C2 [×3], C3 [×4], C4 [×4], C22, C6 [×12], C2×C4, C2×C4 [×2], C32, Dic3 [×12], C12 [×4], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, Dic3⋊C4 [×4], C2×C3⋊Dic3 [×2], C6×C12, C6.Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C2×C3⋊S3, Dic3⋊C4 [×4], C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C6.Dic6
►Regular action on 144 points
(1 63 29 86 112 74)(2 64 30 87 113 75)(3 65 31 88 114 76)(4 66 32 89 115 77)(5 67 33 90 116 78)(6 68 34 91 117 79)(7 69 35 92 118 80)(8 70 36 93 119 81)(9 71 25 94 120 82)(10 72 26 95 109 83)(11 61 27 96 110 84)(12 62 28 85 111 73)(13 42 140 99 127 56)(14 43 141 100 128 57)(15 44 142 101 129 58)(16 45 143 102 130 59)(17 46 144 103 131 60)(18 47 133 104 132 49)(19 48 134 105 121 50)(20 37 135 106 122 51)(21 38 136 107 123 52)(22 39 137 108 124 53)(23 40 138 97 125 54)(24 41 139 98 126 55)
(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)
(1 129 92 38)(2 43 93 122)(3 127 94 48)(4 41 95 132)(5 125 96 46)(6 39 85 130)(7 123 86 44)(8 37 87 128)(9 121 88 42)(10 47 89 126)(11 131 90 40)(12 45 91 124)(13 71 105 114)(14 119 106 64)(15 69 107 112)(16 117 108 62)(17 67 97 110)(18 115 98 72)(19 65 99 120)(20 113 100 70)(21 63 101 118)(22 111 102 68)(23 61 103 116)(24 109 104 66)(25 134 76 56)(26 49 77 139)(27 144 78 54)(28 59 79 137)(29 142 80 52)(30 57 81 135)(31 140 82 50)(32 55 83 133)(33 138 84 60)(34 53 73 143)(35 136 74 58)(36 51 75 141)
G:=sub<Sym(144)| (1,63,29,86,112,74)(2,64,30,87,113,75)(3,65,31,88,114,76)(4,66,32,89,115,77)(5,67,33,90,116,78)(6,68,34,91,117,79)(7,69,35,92,118,80)(8,70,36,93,119,81)(9,71,25,94,120,82)(10,72,26,95,109,83)(11,61,27,96,110,84)(12,62,28,85,111,73)(13,42,140,99,127,56)(14,43,141,100,128,57)(15,44,142,101,129,58)(16,45,143,102,130,59)(17,46,144,103,131,60)(18,47,133,104,132,49)(19,48,134,105,121,50)(20,37,135,106,122,51)(21,38,136,107,123,52)(22,39,137,108,124,53)(23,40,138,97,125,54)(24,41,139,98,126,55), (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), (1,129,92,38)(2,43,93,122)(3,127,94,48)(4,41,95,132)(5,125,96,46)(6,39,85,130)(7,123,86,44)(8,37,87,128)(9,121,88,42)(10,47,89,126)(11,131,90,40)(12,45,91,124)(13,71,105,114)(14,119,106,64)(15,69,107,112)(16,117,108,62)(17,67,97,110)(18,115,98,72)(19,65,99,120)(20,113,100,70)(21,63,101,118)(22,111,102,68)(23,61,103,116)(24,109,104,66)(25,134,76,56)(26,49,77,139)(27,144,78,54)(28,59,79,137)(29,142,80,52)(30,57,81,135)(31,140,82,50)(32,55,83,133)(33,138,84,60)(34,53,73,143)(35,136,74,58)(36,51,75,141)>;
G:=Group( (1,63,29,86,112,74)(2,64,30,87,113,75)(3,65,31,88,114,76)(4,66,32,89,115,77)(5,67,33,90,116,78)(6,68,34,91,117,79)(7,69,35,92,118,80)(8,70,36,93,119,81)(9,71,25,94,120,82)(10,72,26,95,109,83)(11,61,27,96,110,84)(12,62,28,85,111,73)(13,42,140,99,127,56)(14,43,141,100,128,57)(15,44,142,101,129,58)(16,45,143,102,130,59)(17,46,144,103,131,60)(18,47,133,104,132,49)(19,48,134,105,121,50)(20,37,135,106,122,51)(21,38,136,107,123,52)(22,39,137,108,124,53)(23,40,138,97,125,54)(24,41,139,98,126,55), (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), (1,129,92,38)(2,43,93,122)(3,127,94,48)(4,41,95,132)(5,125,96,46)(6,39,85,130)(7,123,86,44)(8,37,87,128)(9,121,88,42)(10,47,89,126)(11,131,90,40)(12,45,91,124)(13,71,105,114)(14,119,106,64)(15,69,107,112)(16,117,108,62)(17,67,97,110)(18,115,98,72)(19,65,99,120)(20,113,100,70)(21,63,101,118)(22,111,102,68)(23,61,103,116)(24,109,104,66)(25,134,76,56)(26,49,77,139)(27,144,78,54)(28,59,79,137)(29,142,80,52)(30,57,81,135)(31,140,82,50)(32,55,83,133)(33,138,84,60)(34,53,73,143)(35,136,74,58)(36,51,75,141) );
G=PermutationGroup([(1,63,29,86,112,74),(2,64,30,87,113,75),(3,65,31,88,114,76),(4,66,32,89,115,77),(5,67,33,90,116,78),(6,68,34,91,117,79),(7,69,35,92,118,80),(8,70,36,93,119,81),(9,71,25,94,120,82),(10,72,26,95,109,83),(11,61,27,96,110,84),(12,62,28,85,111,73),(13,42,140,99,127,56),(14,43,141,100,128,57),(15,44,142,101,129,58),(16,45,143,102,130,59),(17,46,144,103,131,60),(18,47,133,104,132,49),(19,48,134,105,121,50),(20,37,135,106,122,51),(21,38,136,107,123,52),(22,39,137,108,124,53),(23,40,138,97,125,54),(24,41,139,98,126,55)], [(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)], [(1,129,92,38),(2,43,93,122),(3,127,94,48),(4,41,95,132),(5,125,96,46),(6,39,85,130),(7,123,86,44),(8,37,87,128),(9,121,88,42),(10,47,89,126),(11,131,90,40),(12,45,91,124),(13,71,105,114),(14,119,106,64),(15,69,107,112),(16,117,108,62),(17,67,97,110),(18,115,98,72),(19,65,99,120),(20,113,100,70),(21,63,101,118),(22,111,102,68),(23,61,103,116),(24,109,104,66),(25,134,76,56),(26,49,77,139),(27,144,78,54),(28,59,79,137),(29,142,80,52),(30,57,81,135),(31,140,82,50),(32,55,83,133),(33,138,84,60),(34,53,73,143),(35,136,74,58),(36,51,75,141)])
C6.Dic6 is a maximal subgroup of
Dic3⋊5Dic6 C62.9C23 C62.16C23 C62.17C23 D6⋊Dic6 C62.28C23 C62.29C23 C62.37C23 S3×Dic3⋊C4 C62.47C23 C62.49C23 C62.54C23 C62.55C23 D6⋊1Dic6 D6⋊4Dic6 C62.75C23 C4×C32⋊4Q8 C12.25Dic6 C122⋊16C2 C122⋊2C2 C62.221C23 C62⋊6Q8 C62.223C23 C62.225C23 C62.227C23 C62.228C23 C62.231C23 C12⋊2Dic6 C62.233C23 C62.234C23 C4⋊C4×C3⋊S3 C62.238C23 C62.240C23 C62.242C23 C62⋊10Q8 C4×C32⋊7D4 C62.129D4 C62.72D4 C62⋊14D4 C62.259C23 C62.261C23 C62.19D6 C6.Dic18 C62.81D6 C62.82D6 C62.146D6
C6.Dic6 is a maximal quotient of
C12.9Dic6 C12.10Dic6 C12.30Dic6 C62.8Q8 C62.15Q8 C6.Dic18 C62.29D6 C62.81D6 C62.82D6 C62.146D6
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 | D6 | Dic6 | C4×S3 | C3⋊D4 |
| kernel | C6.Dic6 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C2×C12 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 4 | 8 | 8 | 8 |
Matrix representation of C6.Dic6 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 11 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 8 | 5 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,4,0,0,0,0,0,12,0,0,1,1],[8,0,0,0,0,8,0,0,0,0,2,11,0,0,2,4],[0,1,0,0,1,0,0,0,0,0,8,8,0,0,0,5] >;
C6.Dic6 in GAP, Magma, Sage, TeX
C_6.{\rm Dic}_6 % in TeX
G:=Group("C6.Dic6"); // GroupNames label
G:=SmallGroup(144,93);
// by ID
G=gap.SmallGroup(144,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=1,c^2=a^3*b^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations