direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.58D6, (C6×C12).25C4, (C3×C6)⋊8M4(2), (C2×C12).400D6, (C2×C62).15C4, C6⋊2(C4.Dic3), C62.112(C2×C4), (C22×C12).29S3, C12.61(C2×Dic3), (C2×C12).16Dic3, C32⋊15(C2×M4(2)), (C6×C12).310C22, C12.212(C22×S3), (C3×C12).181C23, C32⋊4C8⋊31C22, C23.4(C3⋊Dic3), (C22×C6).18Dic3, C6.32(C22×Dic3), (C2×C6×C12).11C2, C3⋊3(C2×C4.Dic3), C4.41(C22×C3⋊S3), C4.14(C2×C3⋊Dic3), (C3×C12).126(C2×C4), (C2×C32⋊4C8)⋊20C2, (C22×C4).6(C3⋊S3), (C2×C4).6(C3⋊Dic3), (C2×C6).52(C2×Dic3), C2.3(C22×C3⋊Dic3), (C3×C6).121(C22×C4), C22.12(C2×C3⋊Dic3), (C2×C4).83(C2×C3⋊S3), SmallGroup(288,778)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.58D6
G = < a,b,c,d | a2=b12=c6=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 404 in 204 conjugacy classes, 125 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C3⋊C8, C2×C12, C22×C6, C2×M4(2), C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C4.Dic3, C22×C12, C32⋊4C8, C6×C12, C6×C12, C2×C62, C2×C4.Dic3, C2×C32⋊4C8, C12.58D6, C2×C6×C12, C2×C12.58D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C3⋊S3, C2×Dic3, C22×S3, C2×M4(2), C3⋊Dic3, C2×C3⋊S3, C4.Dic3, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, C2×C4.Dic3, C12.58D6, C22×C3⋊Dic3, C2×C12.58D6
►On 144 points
Generators in S144
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 133)(13 58)(14 59)(15 60)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 97)(82 98)(83 99)(84 100)(85 128)(86 129)(87 130)(88 131)(89 132)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)
(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 37 30)(2 38 31)(3 39 32)(4 40 33)(5 41 34)(6 42 35)(7 43 36)(8 44 25)(9 45 26)(10 46 27)(11 47 28)(12 48 29)(13 93 104 19 87 98)(14 94 105 20 88 99)(15 95 106 21 89 100)(16 96 107 22 90 101)(17 85 108 23 91 102)(18 86 97 24 92 103)(49 127 79 55 121 73)(50 128 80 56 122 74)(51 129 81 57 123 75)(52 130 82 58 124 76)(53 131 83 59 125 77)(54 132 84 60 126 78)(61 134 115)(62 135 116)(63 136 117)(64 137 118)(65 138 119)(66 139 120)(67 140 109)(68 141 110)(69 142 111)(70 143 112)(71 144 113)(72 133 114)
(1 81 4 84 7 75 10 78)(2 74 5 77 8 80 11 83)(3 79 6 82 9 73 12 76)(13 63 16 66 19 69 22 72)(14 68 17 71 20 62 23 65)(15 61 18 64 21 67 24 70)(25 50 28 53 31 56 34 59)(26 55 29 58 32 49 35 52)(27 60 30 51 33 54 36 57)(37 123 40 126 43 129 46 132)(38 128 41 131 44 122 47 125)(39 121 42 124 45 127 48 130)(85 119 88 110 91 113 94 116)(86 112 89 115 92 118 95 109)(87 117 90 120 93 111 96 114)(97 137 100 140 103 143 106 134)(98 142 101 133 104 136 107 139)(99 135 102 138 105 141 108 144)
G:=sub<Sym(144)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,133)(13,58)(14,59)(15,60)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,97)(82,98)(83,99)(84,100)(85,128)(86,129)(87,130)(88,131)(89,132)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127), (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), (1,37,30)(2,38,31)(3,39,32)(4,40,33)(5,41,34)(6,42,35)(7,43,36)(8,44,25)(9,45,26)(10,46,27)(11,47,28)(12,48,29)(13,93,104,19,87,98)(14,94,105,20,88,99)(15,95,106,21,89,100)(16,96,107,22,90,101)(17,85,108,23,91,102)(18,86,97,24,92,103)(49,127,79,55,121,73)(50,128,80,56,122,74)(51,129,81,57,123,75)(52,130,82,58,124,76)(53,131,83,59,125,77)(54,132,84,60,126,78)(61,134,115)(62,135,116)(63,136,117)(64,137,118)(65,138,119)(66,139,120)(67,140,109)(68,141,110)(69,142,111)(70,143,112)(71,144,113)(72,133,114), (1,81,4,84,7,75,10,78)(2,74,5,77,8,80,11,83)(3,79,6,82,9,73,12,76)(13,63,16,66,19,69,22,72)(14,68,17,71,20,62,23,65)(15,61,18,64,21,67,24,70)(25,50,28,53,31,56,34,59)(26,55,29,58,32,49,35,52)(27,60,30,51,33,54,36,57)(37,123,40,126,43,129,46,132)(38,128,41,131,44,122,47,125)(39,121,42,124,45,127,48,130)(85,119,88,110,91,113,94,116)(86,112,89,115,92,118,95,109)(87,117,90,120,93,111,96,114)(97,137,100,140,103,143,106,134)(98,142,101,133,104,136,107,139)(99,135,102,138,105,141,108,144)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,133)(13,58)(14,59)(15,60)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,97)(82,98)(83,99)(84,100)(85,128)(86,129)(87,130)(88,131)(89,132)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127), (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), (1,37,30)(2,38,31)(3,39,32)(4,40,33)(5,41,34)(6,42,35)(7,43,36)(8,44,25)(9,45,26)(10,46,27)(11,47,28)(12,48,29)(13,93,104,19,87,98)(14,94,105,20,88,99)(15,95,106,21,89,100)(16,96,107,22,90,101)(17,85,108,23,91,102)(18,86,97,24,92,103)(49,127,79,55,121,73)(50,128,80,56,122,74)(51,129,81,57,123,75)(52,130,82,58,124,76)(53,131,83,59,125,77)(54,132,84,60,126,78)(61,134,115)(62,135,116)(63,136,117)(64,137,118)(65,138,119)(66,139,120)(67,140,109)(68,141,110)(69,142,111)(70,143,112)(71,144,113)(72,133,114), (1,81,4,84,7,75,10,78)(2,74,5,77,8,80,11,83)(3,79,6,82,9,73,12,76)(13,63,16,66,19,69,22,72)(14,68,17,71,20,62,23,65)(15,61,18,64,21,67,24,70)(25,50,28,53,31,56,34,59)(26,55,29,58,32,49,35,52)(27,60,30,51,33,54,36,57)(37,123,40,126,43,129,46,132)(38,128,41,131,44,122,47,125)(39,121,42,124,45,127,48,130)(85,119,88,110,91,113,94,116)(86,112,89,115,92,118,95,109)(87,117,90,120,93,111,96,114)(97,137,100,140,103,143,106,134)(98,142,101,133,104,136,107,139)(99,135,102,138,105,141,108,144) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,133),(13,58),(14,59),(15,60),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,68),(26,69),(27,70),(28,71),(29,72),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,109),(44,110),(45,111),(46,112),(47,113),(48,114),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,97),(82,98),(83,99),(84,100),(85,128),(86,129),(87,130),(88,131),(89,132),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127)], [(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)], [(1,37,30),(2,38,31),(3,39,32),(4,40,33),(5,41,34),(6,42,35),(7,43,36),(8,44,25),(9,45,26),(10,46,27),(11,47,28),(12,48,29),(13,93,104,19,87,98),(14,94,105,20,88,99),(15,95,106,21,89,100),(16,96,107,22,90,101),(17,85,108,23,91,102),(18,86,97,24,92,103),(49,127,79,55,121,73),(50,128,80,56,122,74),(51,129,81,57,123,75),(52,130,82,58,124,76),(53,131,83,59,125,77),(54,132,84,60,126,78),(61,134,115),(62,135,116),(63,136,117),(64,137,118),(65,138,119),(66,139,120),(67,140,109),(68,141,110),(69,142,111),(70,143,112),(71,144,113),(72,133,114)], [(1,81,4,84,7,75,10,78),(2,74,5,77,8,80,11,83),(3,79,6,82,9,73,12,76),(13,63,16,66,19,69,22,72),(14,68,17,71,20,62,23,65),(15,61,18,64,21,67,24,70),(25,50,28,53,31,56,34,59),(26,55,29,58,32,49,35,52),(27,60,30,51,33,54,36,57),(37,123,40,126,43,129,46,132),(38,128,41,131,44,122,47,125),(39,121,42,124,45,127,48,130),(85,119,88,110,91,113,94,116),(86,112,89,115,92,118,95,109),(87,117,90,120,93,111,96,114),(97,137,100,140,103,143,106,134),(98,142,101,133,104,136,107,139),(99,135,102,138,105,141,108,144)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6AB | 8A | ··· | 8H | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 |
| kernel | C2×C12.58D6 | C2×C32⋊4C8 | C12.58D6 | C2×C6×C12 | C6×C12 | C2×C62 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 12 | 12 | 4 | 4 | 32 |
Matrix representation of C2×C12.58D6 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 8 | 24 | 0 | 0 |
| 0 | 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 34 | 72 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 27 | 7 | 0 | 0 |
| 0 | 4 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 0 | 0 | 0 | 70 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,3,8,0,0,0,0,24,0,0,0,0,0,24,0,0,0,0,0,3],[72,0,0,0,0,0,1,34,0,0,0,0,72,0,0,0,0,0,8,0,0,0,0,0,9],[46,0,0,0,0,0,27,4,0,0,0,7,46,0,0,0,0,0,0,70,0,0,0,64,0] >;
C2×C12.58D6 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{58}D_6 % in TeX
G:=Group("C2xC12.58D6"); // GroupNames label
G:=SmallGroup(288,778);
// by ID
G=gap.SmallGroup(288,778);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations