metabelian, supersoluble, monomial
Aliases: C62.262C23, (C6×Q8)⋊8S3, (C2×C12).254D6, (C3×C12).106D4, C12.65(C3⋊D4), C6.11D12⋊27C2, C3⋊4(C12.23D4), (C6×C12).151C22, C6.55(Q8⋊3S3), C4.11(C32⋊7D4), C32⋊18(C4.4D4), C2.9(C12.26D6), (Q8×C3×C6)⋊8C2, (C2×Q8)⋊6(C3⋊S3), (C4×C3⋊Dic3)⋊11C2, (C3×C6).292(C2×D4), C6.133(C2×C3⋊D4), (C2×C12⋊S3).16C2, C2.22(C2×C32⋊7D4), (C3×C6).164(C4○D4), (C2×C6).279(C22×S3), C22.65(C22×C3⋊S3), (C22×C3⋊S3).48C22, (C2×C3⋊Dic3).171C22, (C2×C4).57(C2×C3⋊S3), SmallGroup(288,804)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.262C23
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, dcd-1=b3c, ece-1=a3c, ede-1=b3d >
Subgroups: 972 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C4.4D4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C4×Dic3, D6⋊C4, C2×D12, C6×Q8, C12⋊S3, C2×C3⋊Dic3, C6×C12, C6×C12, Q8×C32, C22×C3⋊S3, C12.23D4, C4×C3⋊Dic3, C6.11D12, C2×C12⋊S3, Q8×C3×C6, C62.262C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C4.4D4, C2×C3⋊S3, Q8⋊3S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C12.23D4, C12.26D6, C2×C32⋊7D4, C62.262C23
►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 35 38 57 13 52)(2 36 39 58 14 53)(3 31 40 59 15 54)(4 32 41 60 16 49)(5 33 42 55 17 50)(6 34 37 56 18 51)(7 134 20 142 120 27)(8 135 21 143 115 28)(9 136 22 144 116 29)(10 137 23 139 117 30)(11 138 24 140 118 25)(12 133 19 141 119 26)(43 78 91 63 86 71)(44 73 92 64 87 72)(45 74 93 65 88 67)(46 75 94 66 89 68)(47 76 95 61 90 69)(48 77 96 62 85 70)(79 114 127 99 122 107)(80 109 128 100 123 108)(81 110 129 101 124 103)(82 111 130 102 125 104)(83 112 131 97 126 105)(84 113 132 98 121 106)
(2 6)(3 5)(7 139)(8 144)(9 143)(10 142)(11 141)(12 140)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 138)(20 137)(21 136)(22 135)(23 134)(24 133)(25 119)(26 118)(27 117)(28 116)(29 115)(30 120)(31 50)(32 49)(33 54)(34 53)(35 52)(36 51)(43 87)(44 86)(45 85)(46 90)(47 89)(48 88)(55 59)(56 58)(61 75)(62 74)(63 73)(64 78)(65 77)(66 76)(67 70)(68 69)(71 72)(79 112)(80 111)(81 110)(82 109)(83 114)(84 113)(91 92)(93 96)(94 95)(97 122)(98 121)(99 126)(100 125)(101 124)(102 123)(103 129)(104 128)(105 127)(106 132)(107 131)(108 130)
(1 129 57 103)(2 130 58 104)(3 131 59 105)(4 132 60 106)(5 127 55 107)(6 128 56 108)(7 96 142 70)(8 91 143 71)(9 92 144 72)(10 93 139 67)(11 94 140 68)(12 95 141 69)(13 81 35 101)(14 82 36 102)(15 83 31 97)(16 84 32 98)(17 79 33 99)(18 80 34 100)(19 90 26 76)(20 85 27 77)(21 86 28 78)(22 87 29 73)(23 88 30 74)(24 89 25 75)(37 123 51 109)(38 124 52 110)(39 125 53 111)(40 126 54 112)(41 121 49 113)(42 122 50 114)(43 135 63 115)(44 136 64 116)(45 137 65 117)(46 138 66 118)(47 133 61 119)(48 134 62 120)
(1 93 57 67)(2 94 58 68)(3 95 59 69)(4 96 60 70)(5 91 55 71)(6 92 56 72)(7 106 142 132)(8 107 143 127)(9 108 144 128)(10 103 139 129)(11 104 140 130)(12 105 141 131)(13 45 35 65)(14 46 36 66)(15 47 31 61)(16 48 32 62)(17 43 33 63)(18 44 34 64)(19 112 26 126)(20 113 27 121)(21 114 28 122)(22 109 29 123)(23 110 30 124)(24 111 25 125)(37 87 51 73)(38 88 52 74)(39 89 53 75)(40 90 54 76)(41 85 49 77)(42 86 50 78)(79 115 99 135)(80 116 100 136)(81 117 101 137)(82 118 102 138)(83 119 97 133)(84 120 98 134)
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), (1,35,38,57,13,52)(2,36,39,58,14,53)(3,31,40,59,15,54)(4,32,41,60,16,49)(5,33,42,55,17,50)(6,34,37,56,18,51)(7,134,20,142,120,27)(8,135,21,143,115,28)(9,136,22,144,116,29)(10,137,23,139,117,30)(11,138,24,140,118,25)(12,133,19,141,119,26)(43,78,91,63,86,71)(44,73,92,64,87,72)(45,74,93,65,88,67)(46,75,94,66,89,68)(47,76,95,61,90,69)(48,77,96,62,85,70)(79,114,127,99,122,107)(80,109,128,100,123,108)(81,110,129,101,124,103)(82,111,130,102,125,104)(83,112,131,97,126,105)(84,113,132,98,121,106), (2,6)(3,5)(7,139)(8,144)(9,143)(10,142)(11,141)(12,140)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,138)(20,137)(21,136)(22,135)(23,134)(24,133)(25,119)(26,118)(27,117)(28,116)(29,115)(30,120)(31,50)(32,49)(33,54)(34,53)(35,52)(36,51)(43,87)(44,86)(45,85)(46,90)(47,89)(48,88)(55,59)(56,58)(61,75)(62,74)(63,73)(64,78)(65,77)(66,76)(67,70)(68,69)(71,72)(79,112)(80,111)(81,110)(82,109)(83,114)(84,113)(91,92)(93,96)(94,95)(97,122)(98,121)(99,126)(100,125)(101,124)(102,123)(103,129)(104,128)(105,127)(106,132)(107,131)(108,130), (1,129,57,103)(2,130,58,104)(3,131,59,105)(4,132,60,106)(5,127,55,107)(6,128,56,108)(7,96,142,70)(8,91,143,71)(9,92,144,72)(10,93,139,67)(11,94,140,68)(12,95,141,69)(13,81,35,101)(14,82,36,102)(15,83,31,97)(16,84,32,98)(17,79,33,99)(18,80,34,100)(19,90,26,76)(20,85,27,77)(21,86,28,78)(22,87,29,73)(23,88,30,74)(24,89,25,75)(37,123,51,109)(38,124,52,110)(39,125,53,111)(40,126,54,112)(41,121,49,113)(42,122,50,114)(43,135,63,115)(44,136,64,116)(45,137,65,117)(46,138,66,118)(47,133,61,119)(48,134,62,120), (1,93,57,67)(2,94,58,68)(3,95,59,69)(4,96,60,70)(5,91,55,71)(6,92,56,72)(7,106,142,132)(8,107,143,127)(9,108,144,128)(10,103,139,129)(11,104,140,130)(12,105,141,131)(13,45,35,65)(14,46,36,66)(15,47,31,61)(16,48,32,62)(17,43,33,63)(18,44,34,64)(19,112,26,126)(20,113,27,121)(21,114,28,122)(22,109,29,123)(23,110,30,124)(24,111,25,125)(37,87,51,73)(38,88,52,74)(39,89,53,75)(40,90,54,76)(41,85,49,77)(42,86,50,78)(79,115,99,135)(80,116,100,136)(81,117,101,137)(82,118,102,138)(83,119,97,133)(84,120,98,134)>;
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), (1,35,38,57,13,52)(2,36,39,58,14,53)(3,31,40,59,15,54)(4,32,41,60,16,49)(5,33,42,55,17,50)(6,34,37,56,18,51)(7,134,20,142,120,27)(8,135,21,143,115,28)(9,136,22,144,116,29)(10,137,23,139,117,30)(11,138,24,140,118,25)(12,133,19,141,119,26)(43,78,91,63,86,71)(44,73,92,64,87,72)(45,74,93,65,88,67)(46,75,94,66,89,68)(47,76,95,61,90,69)(48,77,96,62,85,70)(79,114,127,99,122,107)(80,109,128,100,123,108)(81,110,129,101,124,103)(82,111,130,102,125,104)(83,112,131,97,126,105)(84,113,132,98,121,106), (2,6)(3,5)(7,139)(8,144)(9,143)(10,142)(11,141)(12,140)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,138)(20,137)(21,136)(22,135)(23,134)(24,133)(25,119)(26,118)(27,117)(28,116)(29,115)(30,120)(31,50)(32,49)(33,54)(34,53)(35,52)(36,51)(43,87)(44,86)(45,85)(46,90)(47,89)(48,88)(55,59)(56,58)(61,75)(62,74)(63,73)(64,78)(65,77)(66,76)(67,70)(68,69)(71,72)(79,112)(80,111)(81,110)(82,109)(83,114)(84,113)(91,92)(93,96)(94,95)(97,122)(98,121)(99,126)(100,125)(101,124)(102,123)(103,129)(104,128)(105,127)(106,132)(107,131)(108,130), (1,129,57,103)(2,130,58,104)(3,131,59,105)(4,132,60,106)(5,127,55,107)(6,128,56,108)(7,96,142,70)(8,91,143,71)(9,92,144,72)(10,93,139,67)(11,94,140,68)(12,95,141,69)(13,81,35,101)(14,82,36,102)(15,83,31,97)(16,84,32,98)(17,79,33,99)(18,80,34,100)(19,90,26,76)(20,85,27,77)(21,86,28,78)(22,87,29,73)(23,88,30,74)(24,89,25,75)(37,123,51,109)(38,124,52,110)(39,125,53,111)(40,126,54,112)(41,121,49,113)(42,122,50,114)(43,135,63,115)(44,136,64,116)(45,137,65,117)(46,138,66,118)(47,133,61,119)(48,134,62,120), (1,93,57,67)(2,94,58,68)(3,95,59,69)(4,96,60,70)(5,91,55,71)(6,92,56,72)(7,106,142,132)(8,107,143,127)(9,108,144,128)(10,103,139,129)(11,104,140,130)(12,105,141,131)(13,45,35,65)(14,46,36,66)(15,47,31,61)(16,48,32,62)(17,43,33,63)(18,44,34,64)(19,112,26,126)(20,113,27,121)(21,114,28,122)(22,109,29,123)(23,110,30,124)(24,111,25,125)(37,87,51,73)(38,88,52,74)(39,89,53,75)(40,90,54,76)(41,85,49,77)(42,86,50,78)(79,115,99,135)(80,116,100,136)(81,117,101,137)(82,118,102,138)(83,119,97,133)(84,120,98,134) );
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)], [(1,35,38,57,13,52),(2,36,39,58,14,53),(3,31,40,59,15,54),(4,32,41,60,16,49),(5,33,42,55,17,50),(6,34,37,56,18,51),(7,134,20,142,120,27),(8,135,21,143,115,28),(9,136,22,144,116,29),(10,137,23,139,117,30),(11,138,24,140,118,25),(12,133,19,141,119,26),(43,78,91,63,86,71),(44,73,92,64,87,72),(45,74,93,65,88,67),(46,75,94,66,89,68),(47,76,95,61,90,69),(48,77,96,62,85,70),(79,114,127,99,122,107),(80,109,128,100,123,108),(81,110,129,101,124,103),(82,111,130,102,125,104),(83,112,131,97,126,105),(84,113,132,98,121,106)], [(2,6),(3,5),(7,139),(8,144),(9,143),(10,142),(11,141),(12,140),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,138),(20,137),(21,136),(22,135),(23,134),(24,133),(25,119),(26,118),(27,117),(28,116),(29,115),(30,120),(31,50),(32,49),(33,54),(34,53),(35,52),(36,51),(43,87),(44,86),(45,85),(46,90),(47,89),(48,88),(55,59),(56,58),(61,75),(62,74),(63,73),(64,78),(65,77),(66,76),(67,70),(68,69),(71,72),(79,112),(80,111),(81,110),(82,109),(83,114),(84,113),(91,92),(93,96),(94,95),(97,122),(98,121),(99,126),(100,125),(101,124),(102,123),(103,129),(104,128),(105,127),(106,132),(107,131),(108,130)], [(1,129,57,103),(2,130,58,104),(3,131,59,105),(4,132,60,106),(5,127,55,107),(6,128,56,108),(7,96,142,70),(8,91,143,71),(9,92,144,72),(10,93,139,67),(11,94,140,68),(12,95,141,69),(13,81,35,101),(14,82,36,102),(15,83,31,97),(16,84,32,98),(17,79,33,99),(18,80,34,100),(19,90,26,76),(20,85,27,77),(21,86,28,78),(22,87,29,73),(23,88,30,74),(24,89,25,75),(37,123,51,109),(38,124,52,110),(39,125,53,111),(40,126,54,112),(41,121,49,113),(42,122,50,114),(43,135,63,115),(44,136,64,116),(45,137,65,117),(46,138,66,118),(47,133,61,119),(48,134,62,120)], [(1,93,57,67),(2,94,58,68),(3,95,59,69),(4,96,60,70),(5,91,55,71),(6,92,56,72),(7,106,142,132),(8,107,143,127),(9,108,144,128),(10,103,139,129),(11,104,140,130),(12,105,141,131),(13,45,35,65),(14,46,36,66),(15,47,31,61),(16,48,32,62),(17,43,33,63),(18,44,34,64),(19,112,26,126),(20,113,27,121),(21,114,28,122),(22,109,29,123),(23,110,30,124),(24,111,25,125),(37,87,51,73),(38,88,52,74),(39,89,53,75),(40,90,54,76),(41,85,49,77),(42,86,50,78),(79,115,99,135),(80,116,100,136),(81,117,101,137),(82,118,102,138),(83,119,97,133),(84,120,98,134)]])
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 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | C3⋊D4 | Q8⋊3S3 |
| kernel | C62.262C23 | C4×C3⋊Dic3 | C6.11D12 | C2×C12⋊S3 | Q8×C3×C6 | C6×Q8 | C3×C12 | C2×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 12 | 4 | 16 | 8 |
Matrix representation of C62.262C23 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 10 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 3 |
| 0 | 0 | 0 | 0 | 7 | 2 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,10,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,7,0,0,0,0,3,2],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,5,11,0,0,0,0,0,8] >;
C62.262C23 in GAP, Magma, Sage, TeX
C_6^2._{262}C_2^3 % in TeX
G:=Group("C6^2.262C2^3"); // GroupNames label
G:=SmallGroup(288,804);
// by ID
G=gap.SmallGroup(288,804);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,100,2693,9414]);
// 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,d*c*d^-1=b^3*c,e*c*e^-1=a^3*c,e*d*e^-1=b^3*d>;
// generators/relations