metabelian, supersoluble, monomial
Aliases: C12.17D12, C62.113D4, (C3×C6).36D8, C12⋊S3⋊8C4, C12.26(C4×S3), (C3×C12).49D4, (C2×C12).87D6, C6.22(D6⋊C4), C6.22(D4⋊S3), C3⋊2(C6.D8), (C3×C6).29SD16, C4.9(C12⋊S3), C32⋊9(D4⋊C4), (C6×C12).54C22, C2.2(C32⋊7D8), C6.12(Q8⋊2S3), C2.5(C6.11D12), C2.2(C32⋊11SD16), C22.14(C32⋊7D4), (C3×C4⋊C4)⋊1S3, C4.1(C4×C3⋊S3), C4⋊C4⋊1(C3⋊S3), (C32×C4⋊C4)⋊2C2, (C3×C12).48(C2×C4), (C2×C32⋊4C8)⋊2C2, (C2×C12⋊S3).9C2, (C2×C6).89(C3⋊D4), (C3×C6).53(C22⋊C4), (C2×C4).36(C2×C3⋊S3), SmallGroup(288,284)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.113D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=a3b3c3 >
Subgroups: 772 in 150 conjugacy classes, 59 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3⋊S3, C3×C6, C3⋊C8, D12, C2×C12, C2×C12, C22×S3, D4⋊C4, C3×C12, C3×C12, C2×C3⋊S3, C62, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C32⋊4C8, C12⋊S3, C12⋊S3, C6×C12, C6×C12, C22×C3⋊S3, C6.D8, C2×C32⋊4C8, C32×C4⋊C4, C2×C12⋊S3, C62.113D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C3⋊S3, C4×S3, D12, C3⋊D4, D4⋊C4, C2×C3⋊S3, D6⋊C4, D4⋊S3, Q8⋊2S3, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6.D8, C6.11D12, C32⋊7D8, C32⋊11SD16, C62.113D4
►On 144 points
Generators in S144
(1 139 99 115 71 134)(2 135 72 116 100 140)(3 141 101 117 65 136)(4 129 66 118 102 142)(5 143 103 119 67 130)(6 131 68 120 104 144)(7 137 97 113 69 132)(8 133 70 114 98 138)(9 84 78 127 57 112)(10 105 58 128 79 85)(11 86 80 121 59 106)(12 107 60 122 73 87)(13 88 74 123 61 108)(14 109 62 124 75 81)(15 82 76 125 63 110)(16 111 64 126 77 83)(17 32 38 51 91 43)(18 44 92 52 39 25)(19 26 40 53 93 45)(20 46 94 54 33 27)(21 28 34 55 95 47)(22 48 96 56 35 29)(23 30 36 49 89 41)(24 42 90 50 37 31)
(1 63 22 5 59 18)(2 19 60 6 23 64)(3 57 24 7 61 20)(4 21 62 8 17 58)(9 90 97 13 94 101)(10 102 95 14 98 91)(11 92 99 15 96 103)(12 104 89 16 100 93)(25 134 125 29 130 121)(26 122 131 30 126 135)(27 136 127 31 132 123)(28 124 133 32 128 129)(33 65 78 37 69 74)(34 75 70 38 79 66)(35 67 80 39 71 76)(36 77 72 40 73 68)(41 111 140 45 107 144)(42 137 108 46 141 112)(43 105 142 47 109 138)(44 139 110 48 143 106)(49 83 116 53 87 120)(50 113 88 54 117 84)(51 85 118 55 81 114)(52 115 82 56 119 86)
(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)
(2 114)(3 7)(4 120)(6 118)(8 116)(9 37)(10 41)(11 35)(12 47)(13 33)(14 45)(15 39)(16 43)(17 83)(18 63)(19 81)(20 61)(21 87)(22 59)(23 85)(24 57)(25 110)(26 75)(27 108)(28 73)(29 106)(30 79)(31 112)(32 77)(34 122)(36 128)(38 126)(40 124)(42 127)(44 125)(46 123)(48 121)(49 58)(50 84)(51 64)(52 82)(53 62)(54 88)(55 60)(56 86)(65 97)(66 131)(67 103)(68 129)(69 101)(70 135)(71 99)(72 133)(74 94)(76 92)(78 90)(80 96)(89 105)(91 111)(93 109)(95 107)(98 140)(100 138)(102 144)(104 142)(113 117)(130 143)(132 141)(134 139)(136 137)
G:=sub<Sym(144)| (1,139,99,115,71,134)(2,135,72,116,100,140)(3,141,101,117,65,136)(4,129,66,118,102,142)(5,143,103,119,67,130)(6,131,68,120,104,144)(7,137,97,113,69,132)(8,133,70,114,98,138)(9,84,78,127,57,112)(10,105,58,128,79,85)(11,86,80,121,59,106)(12,107,60,122,73,87)(13,88,74,123,61,108)(14,109,62,124,75,81)(15,82,76,125,63,110)(16,111,64,126,77,83)(17,32,38,51,91,43)(18,44,92,52,39,25)(19,26,40,53,93,45)(20,46,94,54,33,27)(21,28,34,55,95,47)(22,48,96,56,35,29)(23,30,36,49,89,41)(24,42,90,50,37,31), (1,63,22,5,59,18)(2,19,60,6,23,64)(3,57,24,7,61,20)(4,21,62,8,17,58)(9,90,97,13,94,101)(10,102,95,14,98,91)(11,92,99,15,96,103)(12,104,89,16,100,93)(25,134,125,29,130,121)(26,122,131,30,126,135)(27,136,127,31,132,123)(28,124,133,32,128,129)(33,65,78,37,69,74)(34,75,70,38,79,66)(35,67,80,39,71,76)(36,77,72,40,73,68)(41,111,140,45,107,144)(42,137,108,46,141,112)(43,105,142,47,109,138)(44,139,110,48,143,106)(49,83,116,53,87,120)(50,113,88,54,117,84)(51,85,118,55,81,114)(52,115,82,56,119,86), (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), (2,114)(3,7)(4,120)(6,118)(8,116)(9,37)(10,41)(11,35)(12,47)(13,33)(14,45)(15,39)(16,43)(17,83)(18,63)(19,81)(20,61)(21,87)(22,59)(23,85)(24,57)(25,110)(26,75)(27,108)(28,73)(29,106)(30,79)(31,112)(32,77)(34,122)(36,128)(38,126)(40,124)(42,127)(44,125)(46,123)(48,121)(49,58)(50,84)(51,64)(52,82)(53,62)(54,88)(55,60)(56,86)(65,97)(66,131)(67,103)(68,129)(69,101)(70,135)(71,99)(72,133)(74,94)(76,92)(78,90)(80,96)(89,105)(91,111)(93,109)(95,107)(98,140)(100,138)(102,144)(104,142)(113,117)(130,143)(132,141)(134,139)(136,137)>;
G:=Group( (1,139,99,115,71,134)(2,135,72,116,100,140)(3,141,101,117,65,136)(4,129,66,118,102,142)(5,143,103,119,67,130)(6,131,68,120,104,144)(7,137,97,113,69,132)(8,133,70,114,98,138)(9,84,78,127,57,112)(10,105,58,128,79,85)(11,86,80,121,59,106)(12,107,60,122,73,87)(13,88,74,123,61,108)(14,109,62,124,75,81)(15,82,76,125,63,110)(16,111,64,126,77,83)(17,32,38,51,91,43)(18,44,92,52,39,25)(19,26,40,53,93,45)(20,46,94,54,33,27)(21,28,34,55,95,47)(22,48,96,56,35,29)(23,30,36,49,89,41)(24,42,90,50,37,31), (1,63,22,5,59,18)(2,19,60,6,23,64)(3,57,24,7,61,20)(4,21,62,8,17,58)(9,90,97,13,94,101)(10,102,95,14,98,91)(11,92,99,15,96,103)(12,104,89,16,100,93)(25,134,125,29,130,121)(26,122,131,30,126,135)(27,136,127,31,132,123)(28,124,133,32,128,129)(33,65,78,37,69,74)(34,75,70,38,79,66)(35,67,80,39,71,76)(36,77,72,40,73,68)(41,111,140,45,107,144)(42,137,108,46,141,112)(43,105,142,47,109,138)(44,139,110,48,143,106)(49,83,116,53,87,120)(50,113,88,54,117,84)(51,85,118,55,81,114)(52,115,82,56,119,86), (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), (2,114)(3,7)(4,120)(6,118)(8,116)(9,37)(10,41)(11,35)(12,47)(13,33)(14,45)(15,39)(16,43)(17,83)(18,63)(19,81)(20,61)(21,87)(22,59)(23,85)(24,57)(25,110)(26,75)(27,108)(28,73)(29,106)(30,79)(31,112)(32,77)(34,122)(36,128)(38,126)(40,124)(42,127)(44,125)(46,123)(48,121)(49,58)(50,84)(51,64)(52,82)(53,62)(54,88)(55,60)(56,86)(65,97)(66,131)(67,103)(68,129)(69,101)(70,135)(71,99)(72,133)(74,94)(76,92)(78,90)(80,96)(89,105)(91,111)(93,109)(95,107)(98,140)(100,138)(102,144)(104,142)(113,117)(130,143)(132,141)(134,139)(136,137) );
G=PermutationGroup([[(1,139,99,115,71,134),(2,135,72,116,100,140),(3,141,101,117,65,136),(4,129,66,118,102,142),(5,143,103,119,67,130),(6,131,68,120,104,144),(7,137,97,113,69,132),(8,133,70,114,98,138),(9,84,78,127,57,112),(10,105,58,128,79,85),(11,86,80,121,59,106),(12,107,60,122,73,87),(13,88,74,123,61,108),(14,109,62,124,75,81),(15,82,76,125,63,110),(16,111,64,126,77,83),(17,32,38,51,91,43),(18,44,92,52,39,25),(19,26,40,53,93,45),(20,46,94,54,33,27),(21,28,34,55,95,47),(22,48,96,56,35,29),(23,30,36,49,89,41),(24,42,90,50,37,31)], [(1,63,22,5,59,18),(2,19,60,6,23,64),(3,57,24,7,61,20),(4,21,62,8,17,58),(9,90,97,13,94,101),(10,102,95,14,98,91),(11,92,99,15,96,103),(12,104,89,16,100,93),(25,134,125,29,130,121),(26,122,131,30,126,135),(27,136,127,31,132,123),(28,124,133,32,128,129),(33,65,78,37,69,74),(34,75,70,38,79,66),(35,67,80,39,71,76),(36,77,72,40,73,68),(41,111,140,45,107,144),(42,137,108,46,141,112),(43,105,142,47,109,138),(44,139,110,48,143,106),(49,83,116,53,87,120),(50,113,88,54,117,84),(51,85,118,55,81,114),(52,115,82,56,119,86)], [(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)], [(2,114),(3,7),(4,120),(6,118),(8,116),(9,37),(10,41),(11,35),(12,47),(13,33),(14,45),(15,39),(16,43),(17,83),(18,63),(19,81),(20,61),(21,87),(22,59),(23,85),(24,57),(25,110),(26,75),(27,108),(28,73),(29,106),(30,79),(31,112),(32,77),(34,122),(36,128),(38,126),(40,124),(42,127),(44,125),(46,123),(48,121),(49,58),(50,84),(51,64),(52,82),(53,62),(54,88),(55,60),(56,86),(65,97),(66,131),(67,103),(68,129),(69,101),(70,135),(71,99),(72,133),(74,94),(76,92),(78,90),(80,96),(89,105),(91,111),(93,109),(95,107),(98,140),(100,138),(102,144),(104,142),(113,117),(130,143),(132,141),(134,139),(136,137)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | D4⋊S3 | Q8⋊2S3 |
| kernel | C62.113D4 | C2×C32⋊4C8 | C32×C4⋊C4 | C2×C12⋊S3 | C12⋊S3 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 2 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C62.113D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 7 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,7,0,0,0,0,14,66,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C62.113D4 in GAP, Magma, Sage, TeX
C_6^2._{113}D_4 % in TeX
G:=Group("C6^2.113D4"); // GroupNames label
G:=SmallGroup(288,284);
// by ID
G=gap.SmallGroup(288,284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*b^3*c^3>;
// generators/relations