metabelian, supersoluble, monomial
Aliases: C62.78D6, (C6×Dic3)⋊3S3, (C3×C6).41D12, C6.9(S3×Dic3), C3⋊1(D6⋊Dic3), C32⋊8(D6⋊C4), (C32×C6).42D4, C33⋊11(C22⋊C4), C6.16(C12⋊S3), C2.2(C33⋊6D4), C2.1(C33⋊8D4), (C3×C62).8C22, C6.12(C3⋊D12), C6.25(D6⋊S3), C3⋊3(C6.11D12), C6.12(C32⋊7D4), C32⋊9(C6.D4), (C6×C3⋊S3)⋊2C4, (C2×C6).32S32, C6.19(C4×C3⋊S3), (Dic3×C3×C6)⋊3C2, (C3×C6).49(C4×S3), (C2×C3⋊S3)⋊3Dic3, C22.6(S3×C3⋊S3), C2.4(Dic3×C3⋊S3), (C2×C33⋊5C4)⋊2C2, (C22×C3⋊S3).4S3, (C2×Dic3)⋊1(C3⋊S3), (C3×C6).61(C3⋊D4), (C32×C6).39(C2×C4), (C3×C6).52(C2×Dic3), (C2×C6×C3⋊S3).2C2, (C2×C6).14(C2×C3⋊S3), SmallGroup(432,450)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.78D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >
Subgroups: 1336 in 268 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C3⋊S3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C33⋊5C4, C6×C3⋊S3, C6×C3⋊S3, C3×C62, D6⋊Dic3, C6.11D12, Dic3×C3×C6, C2×C33⋊5C4, C2×C6×C3⋊S3, C62.78D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C2×Dic3, C3⋊D4, S32, C2×C3⋊S3, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C4×C3⋊S3, C12⋊S3, C32⋊7D4, S3×C3⋊S3, D6⋊Dic3, C6.11D12, Dic3×C3⋊S3, C33⋊6D4, C33⋊8D4, C62.78D6
►On 144 points
Generators in S144
(1 19 38 7 13 44)(2 20 39 8 14 45)(3 21 40 9 15 46)(4 22 41 10 16 47)(5 23 42 11 17 48)(6 24 43 12 18 37)(25 63 101 31 69 107)(26 64 102 32 70 108)(27 65 103 33 71 97)(28 66 104 34 72 98)(29 67 105 35 61 99)(30 68 106 36 62 100)(49 85 118 55 91 112)(50 86 119 56 92 113)(51 87 120 57 93 114)(52 88 109 58 94 115)(53 89 110 59 95 116)(54 90 111 60 96 117)(73 124 134 79 130 140)(74 125 135 80 131 141)(75 126 136 81 132 142)(76 127 137 82 121 143)(77 128 138 83 122 144)(78 129 139 84 123 133)
(1 121 21 133 42 80)(2 81 43 134 22 122)(3 123 23 135 44 82)(4 83 45 136 24 124)(5 125 13 137 46 84)(6 73 47 138 14 126)(7 127 15 139 48 74)(8 75 37 140 16 128)(9 129 17 141 38 76)(10 77 39 142 18 130)(11 131 19 143 40 78)(12 79 41 144 20 132)(25 60 65 86 105 109)(26 110 106 87 66 49)(27 50 67 88 107 111)(28 112 108 89 68 51)(29 52 69 90 97 113)(30 114 98 91 70 53)(31 54 71 92 99 115)(32 116 100 93 72 55)(33 56 61 94 101 117)(34 118 102 95 62 57)(35 58 63 96 103 119)(36 120 104 85 64 59)
(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 65 139 115)(2 114 140 64)(3 63 141 113)(4 112 142 62)(5 61 143 111)(6 110 144 72)(7 71 133 109)(8 120 134 70)(9 69 135 119)(10 118 136 68)(11 67 137 117)(12 116 138 66)(13 33 78 88)(14 87 79 32)(15 31 80 86)(16 85 81 30)(17 29 82 96)(18 95 83 28)(19 27 84 94)(20 93 73 26)(21 25 74 92)(22 91 75 36)(23 35 76 90)(24 89 77 34)(37 59 122 98)(38 97 123 58)(39 57 124 108)(40 107 125 56)(41 55 126 106)(42 105 127 54)(43 53 128 104)(44 103 129 52)(45 51 130 102)(46 101 131 50)(47 49 132 100)(48 99 121 60)
G:=sub<Sym(144)| (1,19,38,7,13,44)(2,20,39,8,14,45)(3,21,40,9,15,46)(4,22,41,10,16,47)(5,23,42,11,17,48)(6,24,43,12,18,37)(25,63,101,31,69,107)(26,64,102,32,70,108)(27,65,103,33,71,97)(28,66,104,34,72,98)(29,67,105,35,61,99)(30,68,106,36,62,100)(49,85,118,55,91,112)(50,86,119,56,92,113)(51,87,120,57,93,114)(52,88,109,58,94,115)(53,89,110,59,95,116)(54,90,111,60,96,117)(73,124,134,79,130,140)(74,125,135,80,131,141)(75,126,136,81,132,142)(76,127,137,82,121,143)(77,128,138,83,122,144)(78,129,139,84,123,133), (1,121,21,133,42,80)(2,81,43,134,22,122)(3,123,23,135,44,82)(4,83,45,136,24,124)(5,125,13,137,46,84)(6,73,47,138,14,126)(7,127,15,139,48,74)(8,75,37,140,16,128)(9,129,17,141,38,76)(10,77,39,142,18,130)(11,131,19,143,40,78)(12,79,41,144,20,132)(25,60,65,86,105,109)(26,110,106,87,66,49)(27,50,67,88,107,111)(28,112,108,89,68,51)(29,52,69,90,97,113)(30,114,98,91,70,53)(31,54,71,92,99,115)(32,116,100,93,72,55)(33,56,61,94,101,117)(34,118,102,95,62,57)(35,58,63,96,103,119)(36,120,104,85,64,59), (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), (1,65,139,115)(2,114,140,64)(3,63,141,113)(4,112,142,62)(5,61,143,111)(6,110,144,72)(7,71,133,109)(8,120,134,70)(9,69,135,119)(10,118,136,68)(11,67,137,117)(12,116,138,66)(13,33,78,88)(14,87,79,32)(15,31,80,86)(16,85,81,30)(17,29,82,96)(18,95,83,28)(19,27,84,94)(20,93,73,26)(21,25,74,92)(22,91,75,36)(23,35,76,90)(24,89,77,34)(37,59,122,98)(38,97,123,58)(39,57,124,108)(40,107,125,56)(41,55,126,106)(42,105,127,54)(43,53,128,104)(44,103,129,52)(45,51,130,102)(46,101,131,50)(47,49,132,100)(48,99,121,60)>;
G:=Group( (1,19,38,7,13,44)(2,20,39,8,14,45)(3,21,40,9,15,46)(4,22,41,10,16,47)(5,23,42,11,17,48)(6,24,43,12,18,37)(25,63,101,31,69,107)(26,64,102,32,70,108)(27,65,103,33,71,97)(28,66,104,34,72,98)(29,67,105,35,61,99)(30,68,106,36,62,100)(49,85,118,55,91,112)(50,86,119,56,92,113)(51,87,120,57,93,114)(52,88,109,58,94,115)(53,89,110,59,95,116)(54,90,111,60,96,117)(73,124,134,79,130,140)(74,125,135,80,131,141)(75,126,136,81,132,142)(76,127,137,82,121,143)(77,128,138,83,122,144)(78,129,139,84,123,133), (1,121,21,133,42,80)(2,81,43,134,22,122)(3,123,23,135,44,82)(4,83,45,136,24,124)(5,125,13,137,46,84)(6,73,47,138,14,126)(7,127,15,139,48,74)(8,75,37,140,16,128)(9,129,17,141,38,76)(10,77,39,142,18,130)(11,131,19,143,40,78)(12,79,41,144,20,132)(25,60,65,86,105,109)(26,110,106,87,66,49)(27,50,67,88,107,111)(28,112,108,89,68,51)(29,52,69,90,97,113)(30,114,98,91,70,53)(31,54,71,92,99,115)(32,116,100,93,72,55)(33,56,61,94,101,117)(34,118,102,95,62,57)(35,58,63,96,103,119)(36,120,104,85,64,59), (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), (1,65,139,115)(2,114,140,64)(3,63,141,113)(4,112,142,62)(5,61,143,111)(6,110,144,72)(7,71,133,109)(8,120,134,70)(9,69,135,119)(10,118,136,68)(11,67,137,117)(12,116,138,66)(13,33,78,88)(14,87,79,32)(15,31,80,86)(16,85,81,30)(17,29,82,96)(18,95,83,28)(19,27,84,94)(20,93,73,26)(21,25,74,92)(22,91,75,36)(23,35,76,90)(24,89,77,34)(37,59,122,98)(38,97,123,58)(39,57,124,108)(40,107,125,56)(41,55,126,106)(42,105,127,54)(43,53,128,104)(44,103,129,52)(45,51,130,102)(46,101,131,50)(47,49,132,100)(48,99,121,60) );
G=PermutationGroup([[(1,19,38,7,13,44),(2,20,39,8,14,45),(3,21,40,9,15,46),(4,22,41,10,16,47),(5,23,42,11,17,48),(6,24,43,12,18,37),(25,63,101,31,69,107),(26,64,102,32,70,108),(27,65,103,33,71,97),(28,66,104,34,72,98),(29,67,105,35,61,99),(30,68,106,36,62,100),(49,85,118,55,91,112),(50,86,119,56,92,113),(51,87,120,57,93,114),(52,88,109,58,94,115),(53,89,110,59,95,116),(54,90,111,60,96,117),(73,124,134,79,130,140),(74,125,135,80,131,141),(75,126,136,81,132,142),(76,127,137,82,121,143),(77,128,138,83,122,144),(78,129,139,84,123,133)], [(1,121,21,133,42,80),(2,81,43,134,22,122),(3,123,23,135,44,82),(4,83,45,136,24,124),(5,125,13,137,46,84),(6,73,47,138,14,126),(7,127,15,139,48,74),(8,75,37,140,16,128),(9,129,17,141,38,76),(10,77,39,142,18,130),(11,131,19,143,40,78),(12,79,41,144,20,132),(25,60,65,86,105,109),(26,110,106,87,66,49),(27,50,67,88,107,111),(28,112,108,89,68,51),(29,52,69,90,97,113),(30,114,98,91,70,53),(31,54,71,92,99,115),(32,116,100,93,72,55),(33,56,61,94,101,117),(34,118,102,95,62,57),(35,58,63,96,103,119),(36,120,104,85,64,59)], [(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)], [(1,65,139,115),(2,114,140,64),(3,63,141,113),(4,112,142,62),(5,61,143,111),(6,110,144,72),(7,71,133,109),(8,120,134,70),(9,69,135,119),(10,118,136,68),(11,67,137,117),(12,116,138,66),(13,33,78,88),(14,87,79,32),(15,31,80,86),(16,85,81,30),(17,29,82,96),(18,95,83,28),(19,27,84,94),(20,93,73,26),(21,25,74,92),(22,91,75,36),(23,35,76,90),(24,89,77,34),(37,59,122,98),(38,97,123,58),(39,57,124,108),(40,107,125,56),(41,55,126,106),(42,105,127,54),(43,53,128,104),(44,103,129,52),(45,51,130,102),(46,101,131,50),(47,49,132,100),(48,99,121,60)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 6A | ··· | 6O | 6P | ··· | 6AA | 6AB | 6AC | 6AD | 6AE | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 54 | 54 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Dic3 | D6 | C4×S3 | D12 | C3⋊D4 | S32 | S3×Dic3 | D6⋊S3 | C3⋊D12 |
| kernel | C62.78D6 | Dic3×C3×C6 | C2×C33⋊5C4 | C2×C6×C3⋊S3 | C6×C3⋊S3 | C6×Dic3 | C22×C3⋊S3 | C32×C6 | C2×C3⋊S3 | C62 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 5 | 8 | 8 | 12 | 4 | 4 | 4 | 4 |
Matrix representation of C62.78D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C62.78D6 in GAP, Magma, Sage, TeX
C_6^2._{78}D_6 % in TeX
G:=Group("C6^2.78D6"); // GroupNames label
G:=SmallGroup(432,450);
// by ID
G=gap.SmallGroup(432,450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations