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, C3, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, Dic3⋊C4, C2×C3⋊Dic3, C6×C12, C6.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2×C3⋊S3, Dic3⋊C4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C6.Dic6
►Regular action on 144 points
Generators in S144
(1 56 36 132 91 133)(2 57 25 121 92 134)(3 58 26 122 93 135)(4 59 27 123 94 136)(5 60 28 124 95 137)(6 49 29 125 96 138)(7 50 30 126 85 139)(8 51 31 127 86 140)(9 52 32 128 87 141)(10 53 33 129 88 142)(11 54 34 130 89 143)(12 55 35 131 90 144)(13 42 77 98 118 72)(14 43 78 99 119 61)(15 44 79 100 120 62)(16 45 80 101 109 63)(17 46 81 102 110 64)(18 47 82 103 111 65)(19 48 83 104 112 66)(20 37 84 105 113 67)(21 38 73 106 114 68)(22 39 74 107 115 69)(23 40 75 108 116 70)(24 41 76 97 117 71)
(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 78 126 67)(2 72 127 83)(3 76 128 65)(4 70 129 81)(5 74 130 63)(6 68 131 79)(7 84 132 61)(8 66 121 77)(9 82 122 71)(10 64 123 75)(11 80 124 69)(12 62 125 73)(13 31 104 134)(14 139 105 36)(15 29 106 144)(16 137 107 34)(17 27 108 142)(18 135 97 32)(19 25 98 140)(20 133 99 30)(21 35 100 138)(22 143 101 28)(23 33 102 136)(24 141 103 26)(37 91 119 50)(38 55 120 96)(39 89 109 60)(40 53 110 94)(41 87 111 58)(42 51 112 92)(43 85 113 56)(44 49 114 90)(45 95 115 54)(46 59 116 88)(47 93 117 52)(48 57 118 86)
G:=sub<Sym(144)| (1,56,36,132,91,133)(2,57,25,121,92,134)(3,58,26,122,93,135)(4,59,27,123,94,136)(5,60,28,124,95,137)(6,49,29,125,96,138)(7,50,30,126,85,139)(8,51,31,127,86,140)(9,52,32,128,87,141)(10,53,33,129,88,142)(11,54,34,130,89,143)(12,55,35,131,90,144)(13,42,77,98,118,72)(14,43,78,99,119,61)(15,44,79,100,120,62)(16,45,80,101,109,63)(17,46,81,102,110,64)(18,47,82,103,111,65)(19,48,83,104,112,66)(20,37,84,105,113,67)(21,38,73,106,114,68)(22,39,74,107,115,69)(23,40,75,108,116,70)(24,41,76,97,117,71), (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), (1,78,126,67)(2,72,127,83)(3,76,128,65)(4,70,129,81)(5,74,130,63)(6,68,131,79)(7,84,132,61)(8,66,121,77)(9,82,122,71)(10,64,123,75)(11,80,124,69)(12,62,125,73)(13,31,104,134)(14,139,105,36)(15,29,106,144)(16,137,107,34)(17,27,108,142)(18,135,97,32)(19,25,98,140)(20,133,99,30)(21,35,100,138)(22,143,101,28)(23,33,102,136)(24,141,103,26)(37,91,119,50)(38,55,120,96)(39,89,109,60)(40,53,110,94)(41,87,111,58)(42,51,112,92)(43,85,113,56)(44,49,114,90)(45,95,115,54)(46,59,116,88)(47,93,117,52)(48,57,118,86)>;
G:=Group( (1,56,36,132,91,133)(2,57,25,121,92,134)(3,58,26,122,93,135)(4,59,27,123,94,136)(5,60,28,124,95,137)(6,49,29,125,96,138)(7,50,30,126,85,139)(8,51,31,127,86,140)(9,52,32,128,87,141)(10,53,33,129,88,142)(11,54,34,130,89,143)(12,55,35,131,90,144)(13,42,77,98,118,72)(14,43,78,99,119,61)(15,44,79,100,120,62)(16,45,80,101,109,63)(17,46,81,102,110,64)(18,47,82,103,111,65)(19,48,83,104,112,66)(20,37,84,105,113,67)(21,38,73,106,114,68)(22,39,74,107,115,69)(23,40,75,108,116,70)(24,41,76,97,117,71), (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), (1,78,126,67)(2,72,127,83)(3,76,128,65)(4,70,129,81)(5,74,130,63)(6,68,131,79)(7,84,132,61)(8,66,121,77)(9,82,122,71)(10,64,123,75)(11,80,124,69)(12,62,125,73)(13,31,104,134)(14,139,105,36)(15,29,106,144)(16,137,107,34)(17,27,108,142)(18,135,97,32)(19,25,98,140)(20,133,99,30)(21,35,100,138)(22,143,101,28)(23,33,102,136)(24,141,103,26)(37,91,119,50)(38,55,120,96)(39,89,109,60)(40,53,110,94)(41,87,111,58)(42,51,112,92)(43,85,113,56)(44,49,114,90)(45,95,115,54)(46,59,116,88)(47,93,117,52)(48,57,118,86) );
G=PermutationGroup([[(1,56,36,132,91,133),(2,57,25,121,92,134),(3,58,26,122,93,135),(4,59,27,123,94,136),(5,60,28,124,95,137),(6,49,29,125,96,138),(7,50,30,126,85,139),(8,51,31,127,86,140),(9,52,32,128,87,141),(10,53,33,129,88,142),(11,54,34,130,89,143),(12,55,35,131,90,144),(13,42,77,98,118,72),(14,43,78,99,119,61),(15,44,79,100,120,62),(16,45,80,101,109,63),(17,46,81,102,110,64),(18,47,82,103,111,65),(19,48,83,104,112,66),(20,37,84,105,113,67),(21,38,73,106,114,68),(22,39,74,107,115,69),(23,40,75,108,116,70),(24,41,76,97,117,71)], [(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)], [(1,78,126,67),(2,72,127,83),(3,76,128,65),(4,70,129,81),(5,74,130,63),(6,68,131,79),(7,84,132,61),(8,66,121,77),(9,82,122,71),(10,64,123,75),(11,80,124,69),(12,62,125,73),(13,31,104,134),(14,139,105,36),(15,29,106,144),(16,137,107,34),(17,27,108,142),(18,135,97,32),(19,25,98,140),(20,133,99,30),(21,35,100,138),(22,143,101,28),(23,33,102,136),(24,141,103,26),(37,91,119,50),(38,55,120,96),(39,89,109,60),(40,53,110,94),(41,87,111,58),(42,51,112,92),(43,85,113,56),(44,49,114,90),(45,95,115,54),(46,59,116,88),(47,93,117,52),(48,57,118,86)]])
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