metabelian, supersoluble, monomial
Aliases: C62.261C23, (C6×Q8)⋊7S3, C6.50(S3×Q8), C3⋊5(D6⋊3Q8), (C3×C12).105D4, (C2×C12).253D6, C12.64(C3⋊D4), C12⋊Dic3⋊24C2, C32⋊25(C22⋊Q8), (C6×C12).269C22, C6.54(Q8⋊3S3), C6.Dic6⋊27C2, C4.18(C32⋊7D4), C6.11D12.11C2, C2.8(C12.26D6), (Q8×C3×C6)⋊7C2, (C2×C3⋊S3)⋊9Q8, C2.9(Q8×C3⋊S3), (C2×Q8)⋊5(C3⋊S3), (C3×C6).77(C2×Q8), (C3×C6).291(C2×D4), C6.132(C2×C3⋊D4), C2.21(C2×C32⋊7D4), (C3×C6).163(C4○D4), (C2×C6).278(C22×S3), C22.64(C22×C3⋊S3), (C22×C3⋊S3).94C22, (C2×C3⋊Dic3).170C22, (C2×C4×C3⋊S3).8C2, (C2×C4).56(C2×C3⋊S3), SmallGroup(288,803)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.261C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=b3d >
Subgroups: 796 in 222 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, Q8×C32, C22×C3⋊S3, D6⋊3Q8, C6.Dic6, C12⋊Dic3, C6.11D12, C2×C4×C3⋊S3, Q8×C3×C6, C62.261C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C22⋊Q8, C2×C3⋊S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, D6⋊3Q8, Q8×C3⋊S3, C12.26D6, C2×C32⋊7D4, C62.261C23
►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 18 58 31 39 61)(2 13 59 32 40 62)(3 14 60 33 41 63)(4 15 55 34 42 64)(5 16 56 35 37 65)(6 17 57 36 38 66)(7 126 27 24 140 118)(8 121 28 19 141 119)(9 122 29 20 142 120)(10 123 30 21 143 115)(11 124 25 22 144 116)(12 125 26 23 139 117)(43 92 51 73 101 71)(44 93 52 74 102 72)(45 94 53 75 97 67)(46 95 54 76 98 68)(47 96 49 77 99 69)(48 91 50 78 100 70)(79 128 87 109 137 107)(80 129 88 110 138 108)(81 130 89 111 133 103)(82 131 90 112 134 104)(83 132 85 113 135 105)(84 127 86 114 136 106)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 118)(20 117)(21 116)(22 115)(23 120)(24 119)(32 36)(33 35)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 54)(50 53)(51 52)(67 70)(68 69)(71 72)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(79 135)(80 134)(81 133)(82 138)(83 137)(84 136)(85 87)(88 90)(104 108)(105 107)(109 132)(110 131)(111 130)(112 129)(113 128)(114 127)(121 126)(122 125)(123 124)(139 142)(140 141)(143 144)
(1 89 31 103)(2 90 32 104)(3 85 33 105)(4 86 34 106)(5 87 35 107)(6 88 36 108)(7 95 24 98)(8 96 19 99)(9 91 20 100)(10 92 21 101)(11 93 22 102)(12 94 23 97)(13 112 40 82)(14 113 41 83)(15 114 42 84)(16 109 37 79)(17 110 38 80)(18 111 39 81)(25 74 116 44)(26 75 117 45)(27 76 118 46)(28 77 119 47)(29 78 120 48)(30 73 115 43)(49 141 69 121)(50 142 70 122)(51 143 71 123)(52 144 72 124)(53 139 67 125)(54 140 68 126)(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 53 31 67)(2 54 32 68)(3 49 33 69)(4 50 34 70)(5 51 35 71)(6 52 36 72)(7 134 24 131)(8 135 19 132)(9 136 20 127)(10 137 21 128)(11 138 22 129)(12 133 23 130)(13 76 40 46)(14 77 41 47)(15 78 42 48)(16 73 37 43)(17 74 38 44)(18 75 39 45)(25 80 116 110)(26 81 117 111)(27 82 118 112)(28 83 119 113)(29 84 120 114)(30 79 115 109)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 97 61 94)(59 98 62 95)(60 99 63 96)(85 121 105 141)(86 122 106 142)(87 123 107 143)(88 124 108 144)(89 125 103 139)(90 126 104 140)
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), (1,18,58,31,39,61)(2,13,59,32,40,62)(3,14,60,33,41,63)(4,15,55,34,42,64)(5,16,56,35,37,65)(6,17,57,36,38,66)(7,126,27,24,140,118)(8,121,28,19,141,119)(9,122,29,20,142,120)(10,123,30,21,143,115)(11,124,25,22,144,116)(12,125,26,23,139,117)(43,92,51,73,101,71)(44,93,52,74,102,72)(45,94,53,75,97,67)(46,95,54,76,98,68)(47,96,49,77,99,69)(48,91,50,78,100,70)(79,128,87,109,137,107)(80,129,88,110,138,108)(81,130,89,111,133,103)(82,131,90,112,134,104)(83,132,85,113,135,105)(84,127,86,114,136,106), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(32,36)(33,35)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,54)(50,53)(51,52)(67,70)(68,69)(71,72)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,135)(80,134)(81,133)(82,138)(83,137)(84,136)(85,87)(88,90)(104,108)(105,107)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(121,126)(122,125)(123,124)(139,142)(140,141)(143,144), (1,89,31,103)(2,90,32,104)(3,85,33,105)(4,86,34,106)(5,87,35,107)(6,88,36,108)(7,95,24,98)(8,96,19,99)(9,91,20,100)(10,92,21,101)(11,93,22,102)(12,94,23,97)(13,112,40,82)(14,113,41,83)(15,114,42,84)(16,109,37,79)(17,110,38,80)(18,111,39,81)(25,74,116,44)(26,75,117,45)(27,76,118,46)(28,77,119,47)(29,78,120,48)(30,73,115,43)(49,141,69,121)(50,142,70,122)(51,143,71,123)(52,144,72,124)(53,139,67,125)(54,140,68,126)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,53,31,67)(2,54,32,68)(3,49,33,69)(4,50,34,70)(5,51,35,71)(6,52,36,72)(7,134,24,131)(8,135,19,132)(9,136,20,127)(10,137,21,128)(11,138,22,129)(12,133,23,130)(13,76,40,46)(14,77,41,47)(15,78,42,48)(16,73,37,43)(17,74,38,44)(18,75,39,45)(25,80,116,110)(26,81,117,111)(27,82,118,112)(28,83,119,113)(29,84,120,114)(30,79,115,109)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(85,121,105,141)(86,122,106,142)(87,123,107,143)(88,124,108,144)(89,125,103,139)(90,126,104,140)>;
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), (1,18,58,31,39,61)(2,13,59,32,40,62)(3,14,60,33,41,63)(4,15,55,34,42,64)(5,16,56,35,37,65)(6,17,57,36,38,66)(7,126,27,24,140,118)(8,121,28,19,141,119)(9,122,29,20,142,120)(10,123,30,21,143,115)(11,124,25,22,144,116)(12,125,26,23,139,117)(43,92,51,73,101,71)(44,93,52,74,102,72)(45,94,53,75,97,67)(46,95,54,76,98,68)(47,96,49,77,99,69)(48,91,50,78,100,70)(79,128,87,109,137,107)(80,129,88,110,138,108)(81,130,89,111,133,103)(82,131,90,112,134,104)(83,132,85,113,135,105)(84,127,86,114,136,106), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(32,36)(33,35)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,54)(50,53)(51,52)(67,70)(68,69)(71,72)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,135)(80,134)(81,133)(82,138)(83,137)(84,136)(85,87)(88,90)(104,108)(105,107)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(121,126)(122,125)(123,124)(139,142)(140,141)(143,144), (1,89,31,103)(2,90,32,104)(3,85,33,105)(4,86,34,106)(5,87,35,107)(6,88,36,108)(7,95,24,98)(8,96,19,99)(9,91,20,100)(10,92,21,101)(11,93,22,102)(12,94,23,97)(13,112,40,82)(14,113,41,83)(15,114,42,84)(16,109,37,79)(17,110,38,80)(18,111,39,81)(25,74,116,44)(26,75,117,45)(27,76,118,46)(28,77,119,47)(29,78,120,48)(30,73,115,43)(49,141,69,121)(50,142,70,122)(51,143,71,123)(52,144,72,124)(53,139,67,125)(54,140,68,126)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,53,31,67)(2,54,32,68)(3,49,33,69)(4,50,34,70)(5,51,35,71)(6,52,36,72)(7,134,24,131)(8,135,19,132)(9,136,20,127)(10,137,21,128)(11,138,22,129)(12,133,23,130)(13,76,40,46)(14,77,41,47)(15,78,42,48)(16,73,37,43)(17,74,38,44)(18,75,39,45)(25,80,116,110)(26,81,117,111)(27,82,118,112)(28,83,119,113)(29,84,120,114)(30,79,115,109)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(85,121,105,141)(86,122,106,142)(87,123,107,143)(88,124,108,144)(89,125,103,139)(90,126,104,140) );
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)], [(1,18,58,31,39,61),(2,13,59,32,40,62),(3,14,60,33,41,63),(4,15,55,34,42,64),(5,16,56,35,37,65),(6,17,57,36,38,66),(7,126,27,24,140,118),(8,121,28,19,141,119),(9,122,29,20,142,120),(10,123,30,21,143,115),(11,124,25,22,144,116),(12,125,26,23,139,117),(43,92,51,73,101,71),(44,93,52,74,102,72),(45,94,53,75,97,67),(46,95,54,76,98,68),(47,96,49,77,99,69),(48,91,50,78,100,70),(79,128,87,109,137,107),(80,129,88,110,138,108),(81,130,89,111,133,103),(82,131,90,112,134,104),(83,132,85,113,135,105),(84,127,86,114,136,106)], [(2,6),(3,5),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,118),(20,117),(21,116),(22,115),(23,120),(24,119),(32,36),(33,35),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,54),(50,53),(51,52),(67,70),(68,69),(71,72),(73,93),(74,92),(75,91),(76,96),(77,95),(78,94),(79,135),(80,134),(81,133),(82,138),(83,137),(84,136),(85,87),(88,90),(104,108),(105,107),(109,132),(110,131),(111,130),(112,129),(113,128),(114,127),(121,126),(122,125),(123,124),(139,142),(140,141),(143,144)], [(1,89,31,103),(2,90,32,104),(3,85,33,105),(4,86,34,106),(5,87,35,107),(6,88,36,108),(7,95,24,98),(8,96,19,99),(9,91,20,100),(10,92,21,101),(11,93,22,102),(12,94,23,97),(13,112,40,82),(14,113,41,83),(15,114,42,84),(16,109,37,79),(17,110,38,80),(18,111,39,81),(25,74,116,44),(26,75,117,45),(27,76,118,46),(28,77,119,47),(29,78,120,48),(30,73,115,43),(49,141,69,121),(50,142,70,122),(51,143,71,123),(52,144,72,124),(53,139,67,125),(54,140,68,126),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,133,61,130),(59,134,62,131),(60,135,63,132)], [(1,53,31,67),(2,54,32,68),(3,49,33,69),(4,50,34,70),(5,51,35,71),(6,52,36,72),(7,134,24,131),(8,135,19,132),(9,136,20,127),(10,137,21,128),(11,138,22,129),(12,133,23,130),(13,76,40,46),(14,77,41,47),(15,78,42,48),(16,73,37,43),(17,74,38,44),(18,75,39,45),(25,80,116,110),(26,81,117,111),(27,82,118,112),(28,83,119,113),(29,84,120,114),(30,79,115,109),(55,100,64,91),(56,101,65,92),(57,102,66,93),(58,97,61,94),(59,98,62,95),(60,99,63,96),(85,121,105,141),(86,122,106,142),(87,123,107,143),(88,124,108,144),(89,125,103,139),(90,126,104,140)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 12A | ··· | 12X |
| 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 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | C4○D4 | C3⋊D4 | S3×Q8 | Q8⋊3S3 |
| kernel | C62.261C23 | C6.Dic6 | C12⋊Dic3 | C6.11D12 | C2×C4×C3⋊S3 | Q8×C3×C6 | C6×Q8 | C3×C12 | C2×C3⋊S3 | C2×C12 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C62.261C23 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,11,9,0,0,0,0,4,2] >;
C62.261C23 in GAP, Magma, Sage, TeX
C_6^2._{261}C_2^3 % in TeX
G:=Group("C6^2.261C2^3"); // GroupNames label
G:=SmallGroup(288,803);
// by ID
G=gap.SmallGroup(288,803);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=e^2=b^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,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=b^3*d>;
// generators/relations