metabelian, supersoluble, monomial
Aliases: C62⋊6Q8, C62.222C23, (C2×C6)⋊5Dic6, (C2×C12).29D6, C6.105(S3×D4), C3⋊Dic3.63D4, C6.41(C2×Dic6), (C22×C6).84D6, C12⋊Dic3⋊5C2, C62⋊5C4.5C2, (C6×C12).10C22, C6.Dic6⋊4C2, C32⋊17(C22⋊Q8), C6.92(D4⋊2S3), (C2×C62).61C22, C22⋊2(C32⋊4Q8), C3⋊5(Dic3.D4), C2.6(C12.D6), C2.6(D4×C3⋊S3), (C3×C6).55(C2×Q8), (C3×C6).228(C2×D4), (C3×C22⋊C4).5S3, C23.17(C2×C3⋊S3), (C2×C32⋊4Q8)⋊4C2, C22⋊C4.1(C3⋊S3), C2.6(C2×C32⋊4Q8), (C3×C6).142(C4○D4), (C2×C6).239(C22×S3), (C32×C22⋊C4).1C2, C22.39(C22×C3⋊S3), (C22×C3⋊Dic3).10C2, (C2×C3⋊Dic3).78C22, (C2×C4).5(C2×C3⋊S3), SmallGroup(288,735)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62⋊6Q8 |
Generators and relations for C62⋊6Q8
G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 732 in 222 conjugacy classes, 79 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C22×C6, C22⋊Q8, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C2×C62, Dic3.D4, C6.Dic6, C12⋊Dic3, C62⋊5C4, C32×C22⋊C4, C2×C32⋊4Q8, C22×C3⋊Dic3, C62⋊6Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, Dic6, C22×S3, C22⋊Q8, C2×C3⋊S3, C2×Dic6, S3×D4, D4⋊2S3, C32⋊4Q8, C22×C3⋊S3, Dic3.D4, C2×C32⋊4Q8, D4×C3⋊S3, C12.D6, C62⋊6Q8
►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 33 29 36 4 13)(2 31 30 34 5 14)(3 32 28 35 6 15)(7 22 27 16 71 19)(8 23 25 17 72 20)(9 24 26 18 70 21)(10 57 59 68 66 62)(11 55 60 69 64 63)(12 56 58 67 65 61)(37 46 53 40 51 44)(38 47 54 41 49 45)(39 48 52 42 50 43)(73 142 80 76 139 83)(74 143 81 77 140 84)(75 144 82 78 141 79)(85 106 101 88 103 98)(86 107 102 89 104 99)(87 108 97 90 105 100)(91 120 110 94 117 113)(92 115 111 95 118 114)(93 116 112 96 119 109)(121 133 129 124 136 132)(122 134 130 125 137 127)(123 135 131 126 138 128)
(1 77 23 106)(2 75 24 104)(3 73 22 108)(4 143 20 98)(5 141 21 102)(6 139 19 100)(7 87 15 83)(8 85 13 81)(9 89 14 79)(10 118 49 136)(11 116 50 134)(12 120 51 138)(16 90 28 80)(17 88 29 84)(18 86 30 82)(25 101 33 140)(26 99 31 144)(27 97 32 142)(34 78 70 107)(35 76 71 105)(36 74 72 103)(37 123 58 94)(38 121 59 92)(39 125 60 96)(40 126 61 91)(41 124 62 95)(42 122 63 93)(43 130 55 112)(44 128 56 110)(45 132 57 114)(46 135 67 117)(47 133 68 115)(48 137 69 119)(52 127 64 109)(53 131 65 113)(54 129 66 111)
(1 58 23 37)(2 60 24 39)(3 59 22 38)(4 65 20 53)(5 64 21 52)(6 66 19 54)(7 47 15 68)(8 46 13 67)(9 48 14 69)(10 16 49 28)(11 18 50 30)(12 17 51 29)(25 44 33 56)(26 43 31 55)(27 45 32 57)(34 63 70 42)(35 62 71 41)(36 61 72 40)(73 121 108 92)(74 126 103 91)(75 125 104 96)(76 124 105 95)(77 123 106 94)(78 122 107 93)(79 137 89 119)(80 136 90 118)(81 135 85 117)(82 134 86 116)(83 133 87 115)(84 138 88 120)(97 114 142 132)(98 113 143 131)(99 112 144 130)(100 111 139 129)(101 110 140 128)(102 109 141 127)
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), (1,33,29,36,4,13)(2,31,30,34,5,14)(3,32,28,35,6,15)(7,22,27,16,71,19)(8,23,25,17,72,20)(9,24,26,18,70,21)(10,57,59,68,66,62)(11,55,60,69,64,63)(12,56,58,67,65,61)(37,46,53,40,51,44)(38,47,54,41,49,45)(39,48,52,42,50,43)(73,142,80,76,139,83)(74,143,81,77,140,84)(75,144,82,78,141,79)(85,106,101,88,103,98)(86,107,102,89,104,99)(87,108,97,90,105,100)(91,120,110,94,117,113)(92,115,111,95,118,114)(93,116,112,96,119,109)(121,133,129,124,136,132)(122,134,130,125,137,127)(123,135,131,126,138,128), (1,77,23,106)(2,75,24,104)(3,73,22,108)(4,143,20,98)(5,141,21,102)(6,139,19,100)(7,87,15,83)(8,85,13,81)(9,89,14,79)(10,118,49,136)(11,116,50,134)(12,120,51,138)(16,90,28,80)(17,88,29,84)(18,86,30,82)(25,101,33,140)(26,99,31,144)(27,97,32,142)(34,78,70,107)(35,76,71,105)(36,74,72,103)(37,123,58,94)(38,121,59,92)(39,125,60,96)(40,126,61,91)(41,124,62,95)(42,122,63,93)(43,130,55,112)(44,128,56,110)(45,132,57,114)(46,135,67,117)(47,133,68,115)(48,137,69,119)(52,127,64,109)(53,131,65,113)(54,129,66,111), (1,58,23,37)(2,60,24,39)(3,59,22,38)(4,65,20,53)(5,64,21,52)(6,66,19,54)(7,47,15,68)(8,46,13,67)(9,48,14,69)(10,16,49,28)(11,18,50,30)(12,17,51,29)(25,44,33,56)(26,43,31,55)(27,45,32,57)(34,63,70,42)(35,62,71,41)(36,61,72,40)(73,121,108,92)(74,126,103,91)(75,125,104,96)(76,124,105,95)(77,123,106,94)(78,122,107,93)(79,137,89,119)(80,136,90,118)(81,135,85,117)(82,134,86,116)(83,133,87,115)(84,138,88,120)(97,114,142,132)(98,113,143,131)(99,112,144,130)(100,111,139,129)(101,110,140,128)(102,109,141,127)>;
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), (1,33,29,36,4,13)(2,31,30,34,5,14)(3,32,28,35,6,15)(7,22,27,16,71,19)(8,23,25,17,72,20)(9,24,26,18,70,21)(10,57,59,68,66,62)(11,55,60,69,64,63)(12,56,58,67,65,61)(37,46,53,40,51,44)(38,47,54,41,49,45)(39,48,52,42,50,43)(73,142,80,76,139,83)(74,143,81,77,140,84)(75,144,82,78,141,79)(85,106,101,88,103,98)(86,107,102,89,104,99)(87,108,97,90,105,100)(91,120,110,94,117,113)(92,115,111,95,118,114)(93,116,112,96,119,109)(121,133,129,124,136,132)(122,134,130,125,137,127)(123,135,131,126,138,128), (1,77,23,106)(2,75,24,104)(3,73,22,108)(4,143,20,98)(5,141,21,102)(6,139,19,100)(7,87,15,83)(8,85,13,81)(9,89,14,79)(10,118,49,136)(11,116,50,134)(12,120,51,138)(16,90,28,80)(17,88,29,84)(18,86,30,82)(25,101,33,140)(26,99,31,144)(27,97,32,142)(34,78,70,107)(35,76,71,105)(36,74,72,103)(37,123,58,94)(38,121,59,92)(39,125,60,96)(40,126,61,91)(41,124,62,95)(42,122,63,93)(43,130,55,112)(44,128,56,110)(45,132,57,114)(46,135,67,117)(47,133,68,115)(48,137,69,119)(52,127,64,109)(53,131,65,113)(54,129,66,111), (1,58,23,37)(2,60,24,39)(3,59,22,38)(4,65,20,53)(5,64,21,52)(6,66,19,54)(7,47,15,68)(8,46,13,67)(9,48,14,69)(10,16,49,28)(11,18,50,30)(12,17,51,29)(25,44,33,56)(26,43,31,55)(27,45,32,57)(34,63,70,42)(35,62,71,41)(36,61,72,40)(73,121,108,92)(74,126,103,91)(75,125,104,96)(76,124,105,95)(77,123,106,94)(78,122,107,93)(79,137,89,119)(80,136,90,118)(81,135,85,117)(82,134,86,116)(83,133,87,115)(84,138,88,120)(97,114,142,132)(98,113,143,131)(99,112,144,130)(100,111,139,129)(101,110,140,128)(102,109,141,127) );
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)], [(1,33,29,36,4,13),(2,31,30,34,5,14),(3,32,28,35,6,15),(7,22,27,16,71,19),(8,23,25,17,72,20),(9,24,26,18,70,21),(10,57,59,68,66,62),(11,55,60,69,64,63),(12,56,58,67,65,61),(37,46,53,40,51,44),(38,47,54,41,49,45),(39,48,52,42,50,43),(73,142,80,76,139,83),(74,143,81,77,140,84),(75,144,82,78,141,79),(85,106,101,88,103,98),(86,107,102,89,104,99),(87,108,97,90,105,100),(91,120,110,94,117,113),(92,115,111,95,118,114),(93,116,112,96,119,109),(121,133,129,124,136,132),(122,134,130,125,137,127),(123,135,131,126,138,128)], [(1,77,23,106),(2,75,24,104),(3,73,22,108),(4,143,20,98),(5,141,21,102),(6,139,19,100),(7,87,15,83),(8,85,13,81),(9,89,14,79),(10,118,49,136),(11,116,50,134),(12,120,51,138),(16,90,28,80),(17,88,29,84),(18,86,30,82),(25,101,33,140),(26,99,31,144),(27,97,32,142),(34,78,70,107),(35,76,71,105),(36,74,72,103),(37,123,58,94),(38,121,59,92),(39,125,60,96),(40,126,61,91),(41,124,62,95),(42,122,63,93),(43,130,55,112),(44,128,56,110),(45,132,57,114),(46,135,67,117),(47,133,68,115),(48,137,69,119),(52,127,64,109),(53,131,65,113),(54,129,66,111)], [(1,58,23,37),(2,60,24,39),(3,59,22,38),(4,65,20,53),(5,64,21,52),(6,66,19,54),(7,47,15,68),(8,46,13,67),(9,48,14,69),(10,16,49,28),(11,18,50,30),(12,17,51,29),(25,44,33,56),(26,43,31,55),(27,45,32,57),(34,63,70,42),(35,62,71,41),(36,61,72,40),(73,121,108,92),(74,126,103,91),(75,125,104,96),(76,124,105,95),(77,123,106,94),(78,122,107,93),(79,137,89,119),(80,136,90,118),(81,135,85,117),(82,134,86,116),(83,133,87,115),(84,138,88,120),(97,114,142,132),(98,113,143,131),(99,112,144,130),(100,111,139,129),(101,110,140,128),(102,109,141,127)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | Dic6 | S3×D4 | D4⋊2S3 |
| kernel | C62⋊6Q8 | C6.Dic6 | C12⋊Dic3 | C62⋊5C4 | C32×C22⋊C4 | C2×C32⋊4Q8 | C22×C3⋊Dic3 | C3×C22⋊C4 | C3⋊Dic3 | C62 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C62⋊6Q8 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 8 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,5,4,0,0,0,0,0,0,0,8] >;
C62⋊6Q8 in GAP, Magma, Sage, TeX
C_6^2\rtimes_6Q_8
% in TeX
G:=Group("C6^2:6Q8"); // GroupNames label
G:=SmallGroup(288,735);
// by ID
G=gap.SmallGroup(288,735);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,254,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations