metabelian, supersoluble, monomial
Aliases: C62.237C23, C12⋊5(C4×S3), C32⋊21(C4×D4), C3⋊Dic3⋊18D4, C6.114(S3×D4), C12⋊S3⋊11C4, (C2×C12).212D6, C3⋊3(Dic3⋊5D4), C6.11D12⋊23C2, (C6×C12).255C22, C6.49(Q8⋊3S3), C2.2(C12.26D6), C4⋊1(C4×C3⋊S3), (C3×C4⋊C4)⋊4S3, C6.71(S3×C2×C4), C4⋊C4⋊8(C3⋊S3), C2.4(D4×C3⋊S3), (C3×C12)⋊13(C2×C4), (C4×C3⋊Dic3)⋊7C2, (C32×C4⋊C4)⋊13C2, (C3×C6).236(C2×D4), (C2×C12⋊S3).14C2, (C3×C6).160(C4○D4), (C2×C6).254(C22×S3), (C3×C6).102(C22×C4), C22.18(C22×C3⋊S3), (C22×C3⋊S3).85C22, (C2×C3⋊Dic3).185C22, (C2×C4×C3⋊S3)⋊21C2, C2.13(C2×C4×C3⋊S3), (C2×C3⋊S3)⋊11(C2×C4), (C2×C4).43(C2×C3⋊S3), SmallGroup(288,750)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.237C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=b3d >
Subgroups: 1124 in 282 conjugacy classes, 87 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, Dic3⋊5D4, C4×C3⋊Dic3, C6.11D12, C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C62.237C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3, C22×S3, C4×D4, C2×C3⋊S3, S3×C2×C4, S3×D4, Q8⋊3S3, C4×C3⋊S3, C22×C3⋊S3, Dic3⋊5D4, C2×C4×C3⋊S3, D4×C3⋊S3, C12.26D6, C62.237C23
►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 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)
(1 82)(2 81)(3 80)(4 79)(5 84)(6 83)(7 98)(8 97)(9 102)(10 101)(11 100)(12 99)(13 85)(14 90)(15 89)(16 88)(17 87)(18 86)(19 73)(20 78)(21 77)(22 76)(23 75)(24 74)(25 67)(26 72)(27 71)(28 70)(29 69)(30 68)(31 134)(32 133)(33 138)(34 137)(35 136)(36 135)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 125)(44 124)(45 123)(46 122)(47 121)(48 126)(49 119)(50 118)(51 117)(52 116)(53 115)(54 120)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 104)(62 103)(63 108)(64 107)(65 106)(66 105)(91 143)(92 142)(93 141)(94 140)(95 139)(96 144)
(1 86 18 82)(2 87 13 83)(3 88 14 84)(4 89 15 79)(5 90 16 80)(6 85 17 81)(7 74 139 67)(8 75 140 68)(9 76 141 69)(10 77 142 70)(11 78 143 71)(12 73 144 72)(19 99 26 96)(20 100 27 91)(21 101 28 92)(22 102 29 93)(23 97 30 94)(24 98 25 95)(31 110 38 103)(32 111 39 104)(33 112 40 105)(34 113 41 106)(35 114 42 107)(36 109 37 108)(43 125 53 115)(44 126 54 116)(45 121 49 117)(46 122 50 118)(47 123 51 119)(48 124 52 120)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 133 61 130)(59 134 62 131)(60 135 63 132)
(1 43 4 46)(2 44 5 47)(3 45 6 48)(7 103 10 106)(8 104 11 107)(9 105 12 108)(13 54 16 51)(14 49 17 52)(15 50 18 53)(19 132 22 129)(20 127 23 130)(21 128 24 131)(25 134 28 137)(26 135 29 138)(27 136 30 133)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 76 40 73)(38 77 41 74)(39 78 42 75)(55 94 58 91)(56 95 59 92)(57 96 60 93)(61 100 64 97)(62 101 65 98)(63 102 66 99)(79 122 82 125)(80 123 83 126)(81 124 84 121)(85 120 88 117)(86 115 89 118)(87 116 90 119)(109 141 112 144)(110 142 113 139)(111 143 114 140)
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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (1,82)(2,81)(3,80)(4,79)(5,84)(6,83)(7,98)(8,97)(9,102)(10,101)(11,100)(12,99)(13,85)(14,90)(15,89)(16,88)(17,87)(18,86)(19,73)(20,78)(21,77)(22,76)(23,75)(24,74)(25,67)(26,72)(27,71)(28,70)(29,69)(30,68)(31,134)(32,133)(33,138)(34,137)(35,136)(36,135)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,125)(44,124)(45,123)(46,122)(47,121)(48,126)(49,119)(50,118)(51,117)(52,116)(53,115)(54,120)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,104)(62,103)(63,108)(64,107)(65,106)(66,105)(91,143)(92,142)(93,141)(94,140)(95,139)(96,144), (1,86,18,82)(2,87,13,83)(3,88,14,84)(4,89,15,79)(5,90,16,80)(6,85,17,81)(7,74,139,67)(8,75,140,68)(9,76,141,69)(10,77,142,70)(11,78,143,71)(12,73,144,72)(19,99,26,96)(20,100,27,91)(21,101,28,92)(22,102,29,93)(23,97,30,94)(24,98,25,95)(31,110,38,103)(32,111,39,104)(33,112,40,105)(34,113,41,106)(35,114,42,107)(36,109,37,108)(43,125,53,115)(44,126,54,116)(45,121,49,117)(46,122,50,118)(47,123,51,119)(48,124,52,120)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,103,10,106)(8,104,11,107)(9,105,12,108)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,132,22,129)(20,127,23,130)(21,128,24,131)(25,134,28,137)(26,135,29,138)(27,136,30,133)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(109,141,112,144)(110,142,113,139)(111,143,114,140)>;
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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (1,82)(2,81)(3,80)(4,79)(5,84)(6,83)(7,98)(8,97)(9,102)(10,101)(11,100)(12,99)(13,85)(14,90)(15,89)(16,88)(17,87)(18,86)(19,73)(20,78)(21,77)(22,76)(23,75)(24,74)(25,67)(26,72)(27,71)(28,70)(29,69)(30,68)(31,134)(32,133)(33,138)(34,137)(35,136)(36,135)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,125)(44,124)(45,123)(46,122)(47,121)(48,126)(49,119)(50,118)(51,117)(52,116)(53,115)(54,120)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,104)(62,103)(63,108)(64,107)(65,106)(66,105)(91,143)(92,142)(93,141)(94,140)(95,139)(96,144), (1,86,18,82)(2,87,13,83)(3,88,14,84)(4,89,15,79)(5,90,16,80)(6,85,17,81)(7,74,139,67)(8,75,140,68)(9,76,141,69)(10,77,142,70)(11,78,143,71)(12,73,144,72)(19,99,26,96)(20,100,27,91)(21,101,28,92)(22,102,29,93)(23,97,30,94)(24,98,25,95)(31,110,38,103)(32,111,39,104)(33,112,40,105)(34,113,41,106)(35,114,42,107)(36,109,37,108)(43,125,53,115)(44,126,54,116)(45,121,49,117)(46,122,50,118)(47,123,51,119)(48,124,52,120)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,103,10,106)(8,104,11,107)(9,105,12,108)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,132,22,129)(20,127,23,130)(21,128,24,131)(25,134,28,137)(26,135,29,138)(27,136,30,133)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,76,40,73)(38,77,41,74)(39,78,42,75)(55,94,58,91)(56,95,59,92)(57,96,60,93)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,122,82,125)(80,123,83,126)(81,124,84,121)(85,120,88,117)(86,115,89,118)(87,116,90,119)(109,141,112,144)(110,142,113,139)(111,143,114,140) );
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)], [(1,32,58,18,39,61),(2,33,59,13,40,62),(3,34,60,14,41,63),(4,35,55,15,42,64),(5,36,56,16,37,65),(6,31,57,17,38,66),(7,22,117,139,29,121),(8,23,118,140,30,122),(9,24,119,141,25,123),(10,19,120,142,26,124),(11,20,115,143,27,125),(12,21,116,144,28,126),(43,71,91,53,78,100),(44,72,92,54,73,101),(45,67,93,49,74,102),(46,68,94,50,75,97),(47,69,95,51,76,98),(48,70,96,52,77,99),(79,107,127,89,114,136),(80,108,128,90,109,137),(81,103,129,85,110,138),(82,104,130,86,111,133),(83,105,131,87,112,134),(84,106,132,88,113,135)], [(1,82),(2,81),(3,80),(4,79),(5,84),(6,83),(7,98),(8,97),(9,102),(10,101),(11,100),(12,99),(13,85),(14,90),(15,89),(16,88),(17,87),(18,86),(19,73),(20,78),(21,77),(22,76),(23,75),(24,74),(25,67),(26,72),(27,71),(28,70),(29,69),(30,68),(31,134),(32,133),(33,138),(34,137),(35,136),(36,135),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,125),(44,124),(45,123),(46,122),(47,121),(48,126),(49,119),(50,118),(51,117),(52,116),(53,115),(54,120),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109),(61,104),(62,103),(63,108),(64,107),(65,106),(66,105),(91,143),(92,142),(93,141),(94,140),(95,139),(96,144)], [(1,86,18,82),(2,87,13,83),(3,88,14,84),(4,89,15,79),(5,90,16,80),(6,85,17,81),(7,74,139,67),(8,75,140,68),(9,76,141,69),(10,77,142,70),(11,78,143,71),(12,73,144,72),(19,99,26,96),(20,100,27,91),(21,101,28,92),(22,102,29,93),(23,97,30,94),(24,98,25,95),(31,110,38,103),(32,111,39,104),(33,112,40,105),(34,113,41,106),(35,114,42,107),(36,109,37,108),(43,125,53,115),(44,126,54,116),(45,121,49,117),(46,122,50,118),(47,123,51,119),(48,124,52,120),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,133,61,130),(59,134,62,131),(60,135,63,132)], [(1,43,4,46),(2,44,5,47),(3,45,6,48),(7,103,10,106),(8,104,11,107),(9,105,12,108),(13,54,16,51),(14,49,17,52),(15,50,18,53),(19,132,22,129),(20,127,23,130),(21,128,24,131),(25,134,28,137),(26,135,29,138),(27,136,30,133),(31,70,34,67),(32,71,35,68),(33,72,36,69),(37,76,40,73),(38,77,41,74),(39,78,42,75),(55,94,58,91),(56,95,59,92),(57,96,60,93),(61,100,64,97),(62,101,65,98),(63,102,66,99),(79,122,82,125),(80,123,83,126),(81,124,84,121),(85,120,88,117),(86,115,89,118),(87,116,90,119),(109,141,112,144),(110,142,113,139),(111,143,114,140)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 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 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | S3×D4 | Q8⋊3S3 |
| kernel | C62.237C23 | C4×C3⋊Dic3 | C6.11D12 | C32×C4⋊C4 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C12⋊S3 | C3×C4⋊C4 | C3⋊Dic3 | C2×C12 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C62.237C23 ►in GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 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 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 8 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,5,0,0,0,0,8,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C62.237C23 in GAP, Magma, Sage, TeX
C_6^2._{237}C_2^3 % in TeX
G:=Group("C6^2.237C2^3"); // GroupNames label
G:=SmallGroup(288,750);
// by ID
G=gap.SmallGroup(288,750);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=b^3,e^2=a^3,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=b^3*d>;
// generators/relations