metabelian, supersoluble, monomial
Aliases: C62.238C23, C6.115(S3×D4), (C2×C12).213D6, C3⋊5(D6.D4), C6.Dic6⋊6C2, C6.102(C4○D12), C6.11D12⋊24C2, (C6×C12).256C22, C6.50(Q8⋊3S3), C2.5(C12.26D6), C2.14(C12.59D6), C32⋊20(C22.D4), (C3×C4⋊C4)⋊5S3, C4⋊C4⋊2(C3⋊S3), C2.12(D4×C3⋊S3), (C2×C3⋊S3).64D4, (C32×C4⋊C4)⋊14C2, (C3×C6).237(C2×D4), (C2×C12⋊S3).5C2, (C3×C6).117(C4○D4), (C2×C6).255(C22×S3), C22.49(C22×C3⋊S3), (C22×C3⋊S3).86C22, (C2×C3⋊Dic3).85C22, (C2×C4×C3⋊S3)⋊22C2, (C2×C4).11(C2×C3⋊S3), SmallGroup(288,751)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.238C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, 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: 1036 in 234 conjugacy classes, 67 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C22.D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6.D4, C6.Dic6, C6.11D12, C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C62.238C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C22×S3, C22.D4, C2×C3⋊S3, C4○D12, S3×D4, Q8⋊3S3, C22×C3⋊S3, D6.D4, C12.59D6, D4×C3⋊S3, C12.26D6, C62.238C23
►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)
(1 139)(2 144)(3 143)(4 142)(5 141)(6 140)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 33)(20 32)(21 31)(22 36)(23 35)(24 34)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(43 123)(44 122)(45 121)(46 126)(47 125)(48 124)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(61 111)(62 110)(63 109)(64 114)(65 113)(66 112)(67 106)(68 105)(69 104)(70 103)(71 108)(72 107)(73 99)(74 98)(75 97)(76 102)(77 101)(78 100)(79 87)(80 86)(81 85)(82 90)(83 89)(84 88)(91 128)(92 127)(93 132)(94 131)(95 130)(96 129)
(1 106 4 103)(2 107 5 104)(3 108 6 105)(7 96 10 93)(8 91 11 94)(9 92 12 95)(13 98 16 101)(14 99 17 102)(15 100 18 97)(19 90 22 87)(20 85 23 88)(21 86 24 89)(25 75 28 78)(26 76 29 73)(27 77 30 74)(31 80 34 83)(32 81 35 84)(33 82 36 79)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)
(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), (1,139)(2,144)(3,143)(4,142)(5,141)(6,140)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,33)(20,32)(21,31)(22,36)(23,35)(24,34)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,123)(44,122)(45,121)(46,126)(47,125)(48,124)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(61,111)(62,110)(63,109)(64,114)(65,113)(66,112)(67,106)(68,105)(69,104)(70,103)(71,108)(72,107)(73,99)(74,98)(75,97)(76,102)(77,101)(78,100)(79,87)(80,86)(81,85)(82,90)(83,89)(84,88)(91,128)(92,127)(93,132)(94,131)(95,130)(96,129), (1,106,4,103)(2,107,5,104)(3,108,6,105)(7,96,10,93)(8,91,11,94)(9,92,12,95)(13,98,16,101)(14,99,17,102)(15,100,18,97)(19,90,22,87)(20,85,23,88)(21,86,24,89)(25,75,28,78)(26,76,29,73)(27,77,30,74)(31,80,34,83)(32,81,35,84)(33,82,36,79)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141), (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), (1,139)(2,144)(3,143)(4,142)(5,141)(6,140)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,33)(20,32)(21,31)(22,36)(23,35)(24,34)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,123)(44,122)(45,121)(46,126)(47,125)(48,124)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(61,111)(62,110)(63,109)(64,114)(65,113)(66,112)(67,106)(68,105)(69,104)(70,103)(71,108)(72,107)(73,99)(74,98)(75,97)(76,102)(77,101)(78,100)(79,87)(80,86)(81,85)(82,90)(83,89)(84,88)(91,128)(92,127)(93,132)(94,131)(95,130)(96,129), (1,106,4,103)(2,107,5,104)(3,108,6,105)(7,96,10,93)(8,91,11,94)(9,92,12,95)(13,98,16,101)(14,99,17,102)(15,100,18,97)(19,90,22,87)(20,85,23,88)(21,86,24,89)(25,75,28,78)(26,76,29,73)(27,77,30,74)(31,80,34,83)(32,81,35,84)(33,82,36,79)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141), (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)], [(1,139),(2,144),(3,143),(4,142),(5,141),(6,140),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,33),(20,32),(21,31),(22,36),(23,35),(24,34),(37,138),(38,137),(39,136),(40,135),(41,134),(42,133),(43,123),(44,122),(45,121),(46,126),(47,125),(48,124),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(61,111),(62,110),(63,109),(64,114),(65,113),(66,112),(67,106),(68,105),(69,104),(70,103),(71,108),(72,107),(73,99),(74,98),(75,97),(76,102),(77,101),(78,100),(79,87),(80,86),(81,85),(82,90),(83,89),(84,88),(91,128),(92,127),(93,132),(94,131),(95,130),(96,129)], [(1,106,4,103),(2,107,5,104),(3,108,6,105),(7,96,10,93),(8,91,11,94),(9,92,12,95),(13,98,16,101),(14,99,17,102),(15,100,18,97),(19,90,22,87),(20,85,23,88),(21,86,24,89),(25,75,28,78),(26,76,29,73),(27,77,30,74),(31,80,34,83),(32,81,35,84),(33,82,36,79),(37,112,40,109),(38,113,41,110),(39,114,42,111),(43,118,46,115),(44,119,47,116),(45,120,48,117),(49,124,52,121),(50,125,53,122),(51,126,54,123),(55,130,58,127),(56,131,59,128),(57,132,60,129),(61,136,64,133),(62,137,65,134),(63,138,66,135),(67,142,70,139),(68,143,71,140),(69,144,72,141)], [(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 | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C4○D4 | C4○D12 | S3×D4 | Q8⋊3S3 |
| kernel | C62.238C23 | C6.Dic6 | C6.11D12 | C32×C4⋊C4 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C3×C4⋊C4 | C2×C3⋊S3 | C2×C12 | C3×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 4 | 2 | 12 | 4 | 16 | 4 | 4 |
Matrix representation of C62.238C23 ►in GL6(𝔽13)
| 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 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 | 12 |
| 12 | 0 | 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 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 10 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,7],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,10,0,0,0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;
C62.238C23 in GAP, Magma, Sage, TeX
C_6^2._{238}C_2^3 % in TeX
G:=Group("C6^2.238C2^3"); // GroupNames label
G:=SmallGroup(288,751);
// by ID
G=gap.SmallGroup(288,751);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=a^3,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