metabelian, supersoluble, monomial
Aliases: C62.247C23, (C6×C12)⋊13C4, (C2×C12)⋊6Dic3, (C2×C12).429D6, C62.114(C2×C4), C62⋊5C4.8C2, (C22×C12).30S3, C12.62(C2×Dic3), (C22×C6).156D6, C6.106(C4○D12), C12⋊Dic3⋊26C2, (C6×C12).359C22, C6.35(C22×Dic3), C32⋊18(C42⋊C2), C2.4(C12.59D6), C3⋊4(C23.26D6), (C2×C62).108C22, (C2×C6×C12).12C2, (C4×C3⋊Dic3)⋊26C2, (C2×C4)⋊4(C3⋊Dic3), C23.26(C2×C3⋊S3), C4.15(C2×C3⋊Dic3), (C3×C12).139(C2×C4), (C22×C4).9(C3⋊S3), (C2×C6).55(C2×Dic3), C2.5(C22×C3⋊Dic3), C22.5(C2×C3⋊Dic3), (C3×C6).121(C4○D4), (C2×C6).264(C22×S3), (C3×C6).123(C22×C4), C22.22(C22×C3⋊S3), (C2×C3⋊Dic3).163C22, (C2×C4).102(C2×C3⋊S3), SmallGroup(288,783)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.247C23 |
Generators and relations for C62.247C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=a3, d2=b3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, de=ed >
Subgroups: 580 in 228 conjugacy classes, 125 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C2×C3⋊Dic3, C6×C12, C6×C12, C2×C62, C23.26D6, C4×C3⋊Dic3, C12⋊Dic3, C62⋊5C4, C2×C6×C12, C62.247C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C3⋊S3, C2×Dic3, C22×S3, C42⋊C2, C3⋊Dic3, C2×C3⋊S3, C4○D12, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, C23.26D6, C12.59D6, C22×C3⋊Dic3, C62.247C23
►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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 136 123 22 29 142)(8 137 124 23 30 143)(9 138 125 24 25 144)(10 133 126 19 26 139)(11 134 121 20 27 140)(12 135 122 21 28 141)(43 96 67 77 83 50)(44 91 68 78 84 51)(45 92 69 73 79 52)(46 93 70 74 80 53)(47 94 71 75 81 54)(48 95 72 76 82 49)(85 118 131 104 112 98)(86 119 132 105 113 99)(87 120 127 106 114 100)(88 115 128 107 109 101)(89 116 129 108 110 102)(90 117 130 103 111 97)
(1 89 4 86)(2 88 5 85)(3 87 6 90)(7 91 10 94)(8 96 11 93)(9 95 12 92)(13 98 16 101)(14 97 17 100)(15 102 18 99)(19 81 22 84)(20 80 23 83)(21 79 24 82)(25 76 28 73)(26 75 29 78)(27 74 30 77)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 131 40 128)(38 130 41 127)(39 129 42 132)(43 134 46 137)(44 133 47 136)(45 138 48 135)(49 122 52 125)(50 121 53 124)(51 126 54 123)(55 113 58 110)(56 112 59 109)(57 111 60 114)(61 116 64 119)(62 115 65 118)(63 120 66 117)(67 140 70 143)(68 139 71 142)(69 144 72 141)
(1 69 33 52)(2 70 34 53)(3 71 35 54)(4 72 36 49)(5 67 31 50)(6 68 32 51)(7 114 22 120)(8 109 23 115)(9 110 24 116)(10 111 19 117)(11 112 20 118)(12 113 21 119)(13 77 37 43)(14 78 38 44)(15 73 39 45)(16 74 40 46)(17 75 41 47)(18 76 42 48)(25 129 138 102)(26 130 133 97)(27 131 134 98)(28 132 135 99)(29 127 136 100)(30 128 137 101)(55 82 64 95)(56 83 65 96)(57 84 66 91)(58 79 61 92)(59 80 62 93)(60 81 63 94)(85 140 104 121)(86 141 105 122)(87 142 106 123)(88 143 107 124)(89 144 108 125)(90 139 103 126)
(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 138)(26 133)(27 134)(28 135)(29 136)(30 137)(85 104)(86 105)(87 106)(88 107)(89 108)(90 103)(97 130)(98 131)(99 132)(100 127)(101 128)(102 129)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 140)(122 141)(123 142)(124 143)(125 144)(126 139)
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,136,123,22,29,142)(8,137,124,23,30,143)(9,138,125,24,25,144)(10,133,126,19,26,139)(11,134,121,20,27,140)(12,135,122,21,28,141)(43,96,67,77,83,50)(44,91,68,78,84,51)(45,92,69,73,79,52)(46,93,70,74,80,53)(47,94,71,75,81,54)(48,95,72,76,82,49)(85,118,131,104,112,98)(86,119,132,105,113,99)(87,120,127,106,114,100)(88,115,128,107,109,101)(89,116,129,108,110,102)(90,117,130,103,111,97), (1,89,4,86)(2,88,5,85)(3,87,6,90)(7,91,10,94)(8,96,11,93)(9,95,12,92)(13,98,16,101)(14,97,17,100)(15,102,18,99)(19,81,22,84)(20,80,23,83)(21,79,24,82)(25,76,28,73)(26,75,29,78)(27,74,30,77)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,69,33,52)(2,70,34,53)(3,71,35,54)(4,72,36,49)(5,67,31,50)(6,68,32,51)(7,114,22,120)(8,109,23,115)(9,110,24,116)(10,111,19,117)(11,112,20,118)(12,113,21,119)(13,77,37,43)(14,78,38,44)(15,73,39,45)(16,74,40,46)(17,75,41,47)(18,76,42,48)(25,129,138,102)(26,130,133,97)(27,131,134,98)(28,132,135,99)(29,127,136,100)(30,128,137,101)(55,82,64,95)(56,83,65,96)(57,84,66,91)(58,79,61,92)(59,80,62,93)(60,81,63,94)(85,140,104,121)(86,141,105,122)(87,142,106,123)(88,143,107,124)(89,144,108,125)(90,139,103,126), (7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,138)(26,133)(27,134)(28,135)(29,136)(30,137)(85,104)(86,105)(87,106)(88,107)(89,108)(90,103)(97,130)(98,131)(99,132)(100,127)(101,128)(102,129)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,140)(122,141)(123,142)(124,143)(125,144)(126,139)>;
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,136,123,22,29,142)(8,137,124,23,30,143)(9,138,125,24,25,144)(10,133,126,19,26,139)(11,134,121,20,27,140)(12,135,122,21,28,141)(43,96,67,77,83,50)(44,91,68,78,84,51)(45,92,69,73,79,52)(46,93,70,74,80,53)(47,94,71,75,81,54)(48,95,72,76,82,49)(85,118,131,104,112,98)(86,119,132,105,113,99)(87,120,127,106,114,100)(88,115,128,107,109,101)(89,116,129,108,110,102)(90,117,130,103,111,97), (1,89,4,86)(2,88,5,85)(3,87,6,90)(7,91,10,94)(8,96,11,93)(9,95,12,92)(13,98,16,101)(14,97,17,100)(15,102,18,99)(19,81,22,84)(20,80,23,83)(21,79,24,82)(25,76,28,73)(26,75,29,78)(27,74,30,77)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,69,33,52)(2,70,34,53)(3,71,35,54)(4,72,36,49)(5,67,31,50)(6,68,32,51)(7,114,22,120)(8,109,23,115)(9,110,24,116)(10,111,19,117)(11,112,20,118)(12,113,21,119)(13,77,37,43)(14,78,38,44)(15,73,39,45)(16,74,40,46)(17,75,41,47)(18,76,42,48)(25,129,138,102)(26,130,133,97)(27,131,134,98)(28,132,135,99)(29,127,136,100)(30,128,137,101)(55,82,64,95)(56,83,65,96)(57,84,66,91)(58,79,61,92)(59,80,62,93)(60,81,63,94)(85,140,104,121)(86,141,105,122)(87,142,106,123)(88,143,107,124)(89,144,108,125)(90,139,103,126), (7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,138)(26,133)(27,134)(28,135)(29,136)(30,137)(85,104)(86,105)(87,106)(88,107)(89,108)(90,103)(97,130)(98,131)(99,132)(100,127)(101,128)(102,129)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,140)(122,141)(123,142)(124,143)(125,144)(126,139) );
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)], [(1,15,58,33,39,61),(2,16,59,34,40,62),(3,17,60,35,41,63),(4,18,55,36,42,64),(5,13,56,31,37,65),(6,14,57,32,38,66),(7,136,123,22,29,142),(8,137,124,23,30,143),(9,138,125,24,25,144),(10,133,126,19,26,139),(11,134,121,20,27,140),(12,135,122,21,28,141),(43,96,67,77,83,50),(44,91,68,78,84,51),(45,92,69,73,79,52),(46,93,70,74,80,53),(47,94,71,75,81,54),(48,95,72,76,82,49),(85,118,131,104,112,98),(86,119,132,105,113,99),(87,120,127,106,114,100),(88,115,128,107,109,101),(89,116,129,108,110,102),(90,117,130,103,111,97)], [(1,89,4,86),(2,88,5,85),(3,87,6,90),(7,91,10,94),(8,96,11,93),(9,95,12,92),(13,98,16,101),(14,97,17,100),(15,102,18,99),(19,81,22,84),(20,80,23,83),(21,79,24,82),(25,76,28,73),(26,75,29,78),(27,74,30,77),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,131,40,128),(38,130,41,127),(39,129,42,132),(43,134,46,137),(44,133,47,136),(45,138,48,135),(49,122,52,125),(50,121,53,124),(51,126,54,123),(55,113,58,110),(56,112,59,109),(57,111,60,114),(61,116,64,119),(62,115,65,118),(63,120,66,117),(67,140,70,143),(68,139,71,142),(69,144,72,141)], [(1,69,33,52),(2,70,34,53),(3,71,35,54),(4,72,36,49),(5,67,31,50),(6,68,32,51),(7,114,22,120),(8,109,23,115),(9,110,24,116),(10,111,19,117),(11,112,20,118),(12,113,21,119),(13,77,37,43),(14,78,38,44),(15,73,39,45),(16,74,40,46),(17,75,41,47),(18,76,42,48),(25,129,138,102),(26,130,133,97),(27,131,134,98),(28,132,135,99),(29,127,136,100),(30,128,137,101),(55,82,64,95),(56,83,65,96),(57,84,66,91),(58,79,61,92),(59,80,62,93),(60,81,63,94),(85,140,104,121),(86,141,105,122),(87,142,106,123),(88,143,107,124),(89,144,108,125),(90,139,103,126)], [(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(25,138),(26,133),(27,134),(28,135),(29,136),(30,137),(85,104),(86,105),(87,106),(88,107),(89,108),(90,103),(97,130),(98,131),(99,132),(100,127),(101,128),(102,129),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,140),(122,141),(123,142),(124,143),(125,144),(126,139)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C62.247C23 | C4×C3⋊Dic3 | C12⋊Dic3 | C62⋊5C4 | C2×C6×C12 | C6×C12 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 4 | 32 |
Matrix representation of C62.247C23 ►in GL6(𝔽13)
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,2,9,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C62.247C23 in GAP, Magma, Sage, TeX
C_6^2._{247}C_2^3 % in TeX
G:=Group("C6^2.247C2^3"); // GroupNames label
G:=SmallGroup(288,783);
// by ID
G=gap.SmallGroup(288,783);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,422,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=a^3,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations