metabelian, supersoluble, monomial
Aliases: C62⋊10Q8, C62.245C23, (C2×C6)⋊10Dic6, (C2×C12).387D6, (C3×C12).153D4, C6.45(C2×Dic6), (C22×C12).23S3, C62⋊5C4.7C2, C6.Dic6⋊2C2, (C22×C6).154D6, C6.105(C4○D12), C12⋊Dic3⋊11C2, C32⋊23(C22⋊Q8), C3⋊5(C12.48D4), C12.118(C3⋊D4), (C6×C12).303C22, C4.23(C32⋊7D4), C22⋊3(C32⋊4Q8), (C2×C62).106C22, C2.17(C12.59D6), (C2×C6×C12).8C2, (C3×C6).59(C2×Q8), (C3×C6).273(C2×D4), C6.114(C2×C3⋊D4), C23.25(C2×C3⋊S3), (C2×C32⋊4Q8)⋊8C2, C2.5(C2×C32⋊7D4), C2.9(C2×C32⋊4Q8), (C22×C4).7(C3⋊S3), (C3×C6).120(C4○D4), (C2×C6).262(C22×S3), C22.54(C22×C3⋊S3), (C2×C3⋊Dic3).89C22, (C2×C4).84(C2×C3⋊S3), SmallGroup(288,781)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C2×C32⋊4Q8 — C62⋊10Q8 |
Generators and relations for C62⋊10Q8
G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b3, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 668 in 222 conjugacy classes, 89 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊Q8, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C2×C62, C12.48D4, C6.Dic6, C12⋊Dic3, C62⋊5C4, C2×C32⋊4Q8, C2×C6×C12, C62⋊10Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C2×C3⋊S3, C2×Dic6, C4○D12, C2×C3⋊D4, C32⋊4Q8, C32⋊7D4, C22×C3⋊S3, C12.48D4, C2×C32⋊4Q8, C12.59D6, C2×C32⋊7D4, C62⋊10Q8
►On 144 points
Generators in S144
(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 11 35 14 39 41)(2 12 36 15 37 42)(3 10 34 13 38 40)(4 49 16 29 8 45)(5 50 17 30 9 43)(6 51 18 28 7 44)(19 59 25 46 68 52)(20 60 26 47 69 53)(21 58 27 48 67 54)(22 71 57 33 63 65)(23 72 55 31 61 66)(24 70 56 32 62 64)(73 116 84 76 119 81)(74 117 79 77 120 82)(75 118 80 78 115 83)(85 121 99 88 124 102)(86 122 100 89 125 97)(87 123 101 90 126 98)(91 129 112 94 132 109)(92 130 113 95 127 110)(93 131 114 96 128 111)(103 140 138 106 143 135)(104 141 133 107 144 136)(105 142 134 108 139 137)
(1 70 44 26)(2 71 45 27)(3 72 43 25)(4 48 12 57)(5 46 10 55)(6 47 11 56)(7 60 41 24)(8 58 42 22)(9 59 40 23)(13 61 17 52)(14 62 18 53)(15 63 16 54)(19 38 66 30)(20 39 64 28)(21 37 65 29)(31 50 68 34)(32 51 69 35)(33 49 67 36)(73 141 92 122)(74 142 93 123)(75 143 94 124)(76 144 95 125)(77 139 96 126)(78 140 91 121)(79 108 114 90)(80 103 109 85)(81 104 110 86)(82 105 111 87)(83 106 112 88)(84 107 113 89)(97 119 136 127)(98 120 137 128)(99 115 138 129)(100 116 133 130)(101 117 134 131)(102 118 135 132)
(1 78 44 91)(2 74 45 93)(3 76 43 95)(4 111 12 82)(5 113 10 84)(6 109 11 80)(7 129 41 115)(8 131 42 117)(9 127 40 119)(13 73 17 92)(14 75 18 94)(15 77 16 96)(19 104 66 86)(20 106 64 88)(21 108 65 90)(22 101 58 134)(23 97 59 136)(24 99 60 138)(25 144 72 125)(26 140 70 121)(27 142 71 123)(28 112 39 83)(29 114 37 79)(30 110 38 81)(31 100 68 133)(32 102 69 135)(33 98 67 137)(34 116 50 130)(35 118 51 132)(36 120 49 128)(46 107 55 89)(47 103 56 85)(48 105 57 87)(52 141 61 122)(53 143 62 124)(54 139 63 126)
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), (1,11,35,14,39,41)(2,12,36,15,37,42)(3,10,34,13,38,40)(4,49,16,29,8,45)(5,50,17,30,9,43)(6,51,18,28,7,44)(19,59,25,46,68,52)(20,60,26,47,69,53)(21,58,27,48,67,54)(22,71,57,33,63,65)(23,72,55,31,61,66)(24,70,56,32,62,64)(73,116,84,76,119,81)(74,117,79,77,120,82)(75,118,80,78,115,83)(85,121,99,88,124,102)(86,122,100,89,125,97)(87,123,101,90,126,98)(91,129,112,94,132,109)(92,130,113,95,127,110)(93,131,114,96,128,111)(103,140,138,106,143,135)(104,141,133,107,144,136)(105,142,134,108,139,137), (1,70,44,26)(2,71,45,27)(3,72,43,25)(4,48,12,57)(5,46,10,55)(6,47,11,56)(7,60,41,24)(8,58,42,22)(9,59,40,23)(13,61,17,52)(14,62,18,53)(15,63,16,54)(19,38,66,30)(20,39,64,28)(21,37,65,29)(31,50,68,34)(32,51,69,35)(33,49,67,36)(73,141,92,122)(74,142,93,123)(75,143,94,124)(76,144,95,125)(77,139,96,126)(78,140,91,121)(79,108,114,90)(80,103,109,85)(81,104,110,86)(82,105,111,87)(83,106,112,88)(84,107,113,89)(97,119,136,127)(98,120,137,128)(99,115,138,129)(100,116,133,130)(101,117,134,131)(102,118,135,132), (1,78,44,91)(2,74,45,93)(3,76,43,95)(4,111,12,82)(5,113,10,84)(6,109,11,80)(7,129,41,115)(8,131,42,117)(9,127,40,119)(13,73,17,92)(14,75,18,94)(15,77,16,96)(19,104,66,86)(20,106,64,88)(21,108,65,90)(22,101,58,134)(23,97,59,136)(24,99,60,138)(25,144,72,125)(26,140,70,121)(27,142,71,123)(28,112,39,83)(29,114,37,79)(30,110,38,81)(31,100,68,133)(32,102,69,135)(33,98,67,137)(34,116,50,130)(35,118,51,132)(36,120,49,128)(46,107,55,89)(47,103,56,85)(48,105,57,87)(52,141,61,122)(53,143,62,124)(54,139,63,126)>;
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), (1,11,35,14,39,41)(2,12,36,15,37,42)(3,10,34,13,38,40)(4,49,16,29,8,45)(5,50,17,30,9,43)(6,51,18,28,7,44)(19,59,25,46,68,52)(20,60,26,47,69,53)(21,58,27,48,67,54)(22,71,57,33,63,65)(23,72,55,31,61,66)(24,70,56,32,62,64)(73,116,84,76,119,81)(74,117,79,77,120,82)(75,118,80,78,115,83)(85,121,99,88,124,102)(86,122,100,89,125,97)(87,123,101,90,126,98)(91,129,112,94,132,109)(92,130,113,95,127,110)(93,131,114,96,128,111)(103,140,138,106,143,135)(104,141,133,107,144,136)(105,142,134,108,139,137), (1,70,44,26)(2,71,45,27)(3,72,43,25)(4,48,12,57)(5,46,10,55)(6,47,11,56)(7,60,41,24)(8,58,42,22)(9,59,40,23)(13,61,17,52)(14,62,18,53)(15,63,16,54)(19,38,66,30)(20,39,64,28)(21,37,65,29)(31,50,68,34)(32,51,69,35)(33,49,67,36)(73,141,92,122)(74,142,93,123)(75,143,94,124)(76,144,95,125)(77,139,96,126)(78,140,91,121)(79,108,114,90)(80,103,109,85)(81,104,110,86)(82,105,111,87)(83,106,112,88)(84,107,113,89)(97,119,136,127)(98,120,137,128)(99,115,138,129)(100,116,133,130)(101,117,134,131)(102,118,135,132), (1,78,44,91)(2,74,45,93)(3,76,43,95)(4,111,12,82)(5,113,10,84)(6,109,11,80)(7,129,41,115)(8,131,42,117)(9,127,40,119)(13,73,17,92)(14,75,18,94)(15,77,16,96)(19,104,66,86)(20,106,64,88)(21,108,65,90)(22,101,58,134)(23,97,59,136)(24,99,60,138)(25,144,72,125)(26,140,70,121)(27,142,71,123)(28,112,39,83)(29,114,37,79)(30,110,38,81)(31,100,68,133)(32,102,69,135)(33,98,67,137)(34,116,50,130)(35,118,51,132)(36,120,49,128)(46,107,55,89)(47,103,56,85)(48,105,57,87)(52,141,61,122)(53,143,62,124)(54,139,63,126) );
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)], [(1,11,35,14,39,41),(2,12,36,15,37,42),(3,10,34,13,38,40),(4,49,16,29,8,45),(5,50,17,30,9,43),(6,51,18,28,7,44),(19,59,25,46,68,52),(20,60,26,47,69,53),(21,58,27,48,67,54),(22,71,57,33,63,65),(23,72,55,31,61,66),(24,70,56,32,62,64),(73,116,84,76,119,81),(74,117,79,77,120,82),(75,118,80,78,115,83),(85,121,99,88,124,102),(86,122,100,89,125,97),(87,123,101,90,126,98),(91,129,112,94,132,109),(92,130,113,95,127,110),(93,131,114,96,128,111),(103,140,138,106,143,135),(104,141,133,107,144,136),(105,142,134,108,139,137)], [(1,70,44,26),(2,71,45,27),(3,72,43,25),(4,48,12,57),(5,46,10,55),(6,47,11,56),(7,60,41,24),(8,58,42,22),(9,59,40,23),(13,61,17,52),(14,62,18,53),(15,63,16,54),(19,38,66,30),(20,39,64,28),(21,37,65,29),(31,50,68,34),(32,51,69,35),(33,49,67,36),(73,141,92,122),(74,142,93,123),(75,143,94,124),(76,144,95,125),(77,139,96,126),(78,140,91,121),(79,108,114,90),(80,103,109,85),(81,104,110,86),(82,105,111,87),(83,106,112,88),(84,107,113,89),(97,119,136,127),(98,120,137,128),(99,115,138,129),(100,116,133,130),(101,117,134,131),(102,118,135,132)], [(1,78,44,91),(2,74,45,93),(3,76,43,95),(4,111,12,82),(5,113,10,84),(6,109,11,80),(7,129,41,115),(8,131,42,117),(9,127,40,119),(13,73,17,92),(14,75,18,94),(15,77,16,96),(19,104,66,86),(20,106,64,88),(21,108,65,90),(22,101,58,134),(23,97,59,136),(24,99,60,138),(25,144,72,125),(26,140,70,121),(27,142,71,123),(28,112,39,83),(29,114,37,79),(30,110,38,81),(31,100,68,133),(32,102,69,135),(33,98,67,137),(34,116,50,130),(35,118,51,132),(36,120,49,128),(46,107,55,89),(47,103,56,85),(48,105,57,87),(52,141,61,122),(53,143,62,124),(54,139,63,126)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 |
| kernel | C62⋊10Q8 | C6.Dic6 | C12⋊Dic3 | C62⋊5C4 | C2×C32⋊4Q8 | C2×C6×C12 | C22×C12 | C3×C12 | C62 | C2×C12 | C22×C6 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 2 | 16 | 16 | 16 |
Matrix representation of C62⋊10Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C62⋊10Q8 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{10}Q_8 % in TeX
G:=Group("C6^2:10Q8"); // GroupNames label
G:=SmallGroup(288,781);
// by ID
G=gap.SmallGroup(288,781);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations