direct product, metabelian, supersoluble, monomial
Aliases: C2×C62⋊5C4, C62.136D4, C62.263C23, (C2×C62)⋊9C4, C62⋊24(C2×C4), C24.2(C3⋊S3), (C23×C6).12S3, (C22×C6)⋊6Dic3, C6⋊2(C6.D4), C23⋊3(C3⋊Dic3), (C22×C6).161D6, (C22×C62).3C2, C6.39(C22×Dic3), (C2×C62).113C22, C22.25(C32⋊7D4), (C3×C6)⋊8(C22⋊C4), (C2×C6)⋊11(C2×Dic3), (C3×C6).296(C2×D4), C3⋊3(C2×C6.D4), C6.137(C2×C3⋊D4), C23.30(C2×C3⋊S3), C32⋊14(C2×C22⋊C4), C22⋊3(C2×C3⋊Dic3), C2.4(C2×C32⋊7D4), C2.9(C22×C3⋊Dic3), (C2×C6).104(C3⋊D4), (C3×C6).127(C22×C4), (C2×C6).280(C22×S3), (C2×C3⋊Dic3)⋊19C22, (C22×C3⋊Dic3)⋊11C2, C22.27(C22×C3⋊S3), SmallGroup(288,809)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C62⋊5C4 |
Generators and relations for C2×C62⋊5C4
G = < a,b,c,d | a2=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c3, dcd-1=c-1 >
Subgroups: 996 in 396 conjugacy classes, 173 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C23, C23, C23, C32, Dic3, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×C6, C3×C6, C3×C6, C2×Dic3, C22×C6, C22×C6, C2×C22⋊C4, C3⋊Dic3, C62, C62, C62, C6.D4, C22×Dic3, C23×C6, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C2×C62, C2×C6.D4, C62⋊5C4, C22×C3⋊Dic3, C22×C62, C2×C62⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C3⋊S3, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C6.D4, C62⋊5C4, C22×C3⋊Dic3, C2×C32⋊7D4, C2×C62⋊5C4
►On 144 points
Generators in S144
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 10)(8 11)(9 12)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 114)(20 109)(21 110)(22 111)(23 112)(24 113)(25 62)(26 63)(27 64)(28 65)(29 66)(30 61)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(73 76)(74 77)(75 78)(79 86)(80 87)(81 88)(82 89)(83 90)(84 85)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)(103 106)(104 107)(105 108)(115 118)(116 119)(117 120)(121 124)(122 125)(123 126)(127 130)(128 131)(129 132)(133 136)(134 137)(135 138)(139 142)(140 143)(141 144)
(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 65 35 56 25 67)(2 66 36 57 26 68)(3 61 31 58 27 69)(4 62 32 59 28 70)(5 63 33 60 29 71)(6 64 34 55 30 72)(7 47 105 121 42 77)(8 48 106 122 37 78)(9 43 107 123 38 73)(10 44 108 124 39 74)(11 45 103 125 40 75)(12 46 104 126 41 76)(13 110 83 52 24 87)(14 111 84 53 19 88)(15 112 79 54 20 89)(16 113 80 49 21 90)(17 114 81 50 22 85)(18 109 82 51 23 86)(91 137 129 98 141 118)(92 138 130 99 142 119)(93 133 131 100 143 120)(94 134 132 101 144 115)(95 135 127 102 139 116)(96 136 128 97 140 117)
(1 10 51 93)(2 123 52 99)(3 8 53 91)(4 121 54 97)(5 12 49 95)(6 125 50 101)(7 15 96 59)(9 13 92 57)(11 17 94 55)(14 98 58 122)(16 102 60 126)(18 100 56 124)(19 118 61 78)(20 128 62 105)(21 116 63 76)(22 132 64 103)(23 120 65 74)(24 130 66 107)(25 108 109 131)(26 73 110 119)(27 106 111 129)(28 77 112 117)(29 104 113 127)(30 75 114 115)(31 37 88 141)(32 47 89 136)(33 41 90 139)(34 45 85 134)(35 39 86 143)(36 43 87 138)(38 83 142 68)(40 81 144 72)(42 79 140 70)(44 82 133 67)(46 80 135 71)(48 84 137 69)
G:=sub<Sym(144)| (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,10)(8,11)(9,12)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,114)(20,109)(21,110)(22,111)(23,112)(24,113)(25,62)(26,63)(27,64)(28,65)(29,66)(30,61)(31,72)(32,67)(33,68)(34,69)(35,70)(36,71)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(73,76)(74,77)(75,78)(79,86)(80,87)(81,88)(82,89)(83,90)(84,85)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108)(115,118)(116,119)(117,120)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144), (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), (1,65,35,56,25,67)(2,66,36,57,26,68)(3,61,31,58,27,69)(4,62,32,59,28,70)(5,63,33,60,29,71)(6,64,34,55,30,72)(7,47,105,121,42,77)(8,48,106,122,37,78)(9,43,107,123,38,73)(10,44,108,124,39,74)(11,45,103,125,40,75)(12,46,104,126,41,76)(13,110,83,52,24,87)(14,111,84,53,19,88)(15,112,79,54,20,89)(16,113,80,49,21,90)(17,114,81,50,22,85)(18,109,82,51,23,86)(91,137,129,98,141,118)(92,138,130,99,142,119)(93,133,131,100,143,120)(94,134,132,101,144,115)(95,135,127,102,139,116)(96,136,128,97,140,117), (1,10,51,93)(2,123,52,99)(3,8,53,91)(4,121,54,97)(5,12,49,95)(6,125,50,101)(7,15,96,59)(9,13,92,57)(11,17,94,55)(14,98,58,122)(16,102,60,126)(18,100,56,124)(19,118,61,78)(20,128,62,105)(21,116,63,76)(22,132,64,103)(23,120,65,74)(24,130,66,107)(25,108,109,131)(26,73,110,119)(27,106,111,129)(28,77,112,117)(29,104,113,127)(30,75,114,115)(31,37,88,141)(32,47,89,136)(33,41,90,139)(34,45,85,134)(35,39,86,143)(36,43,87,138)(38,83,142,68)(40,81,144,72)(42,79,140,70)(44,82,133,67)(46,80,135,71)(48,84,137,69)>;
G:=Group( (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,10)(8,11)(9,12)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,114)(20,109)(21,110)(22,111)(23,112)(24,113)(25,62)(26,63)(27,64)(28,65)(29,66)(30,61)(31,72)(32,67)(33,68)(34,69)(35,70)(36,71)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(73,76)(74,77)(75,78)(79,86)(80,87)(81,88)(82,89)(83,90)(84,85)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108)(115,118)(116,119)(117,120)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144), (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), (1,65,35,56,25,67)(2,66,36,57,26,68)(3,61,31,58,27,69)(4,62,32,59,28,70)(5,63,33,60,29,71)(6,64,34,55,30,72)(7,47,105,121,42,77)(8,48,106,122,37,78)(9,43,107,123,38,73)(10,44,108,124,39,74)(11,45,103,125,40,75)(12,46,104,126,41,76)(13,110,83,52,24,87)(14,111,84,53,19,88)(15,112,79,54,20,89)(16,113,80,49,21,90)(17,114,81,50,22,85)(18,109,82,51,23,86)(91,137,129,98,141,118)(92,138,130,99,142,119)(93,133,131,100,143,120)(94,134,132,101,144,115)(95,135,127,102,139,116)(96,136,128,97,140,117), (1,10,51,93)(2,123,52,99)(3,8,53,91)(4,121,54,97)(5,12,49,95)(6,125,50,101)(7,15,96,59)(9,13,92,57)(11,17,94,55)(14,98,58,122)(16,102,60,126)(18,100,56,124)(19,118,61,78)(20,128,62,105)(21,116,63,76)(22,132,64,103)(23,120,65,74)(24,130,66,107)(25,108,109,131)(26,73,110,119)(27,106,111,129)(28,77,112,117)(29,104,113,127)(30,75,114,115)(31,37,88,141)(32,47,89,136)(33,41,90,139)(34,45,85,134)(35,39,86,143)(36,43,87,138)(38,83,142,68)(40,81,144,72)(42,79,140,70)(44,82,133,67)(46,80,135,71)(48,84,137,69) );
G=PermutationGroup([[(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,10),(8,11),(9,12),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,114),(20,109),(21,110),(22,111),(23,112),(24,113),(25,62),(26,63),(27,64),(28,65),(29,66),(30,61),(31,72),(32,67),(33,68),(34,69),(35,70),(36,71),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(73,76),(74,77),(75,78),(79,86),(80,87),(81,88),(82,89),(83,90),(84,85),(91,94),(92,95),(93,96),(97,100),(98,101),(99,102),(103,106),(104,107),(105,108),(115,118),(116,119),(117,120),(121,124),(122,125),(123,126),(127,130),(128,131),(129,132),(133,136),(134,137),(135,138),(139,142),(140,143),(141,144)], [(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)], [(1,65,35,56,25,67),(2,66,36,57,26,68),(3,61,31,58,27,69),(4,62,32,59,28,70),(5,63,33,60,29,71),(6,64,34,55,30,72),(7,47,105,121,42,77),(8,48,106,122,37,78),(9,43,107,123,38,73),(10,44,108,124,39,74),(11,45,103,125,40,75),(12,46,104,126,41,76),(13,110,83,52,24,87),(14,111,84,53,19,88),(15,112,79,54,20,89),(16,113,80,49,21,90),(17,114,81,50,22,85),(18,109,82,51,23,86),(91,137,129,98,141,118),(92,138,130,99,142,119),(93,133,131,100,143,120),(94,134,132,101,144,115),(95,135,127,102,139,116),(96,136,128,97,140,117)], [(1,10,51,93),(2,123,52,99),(3,8,53,91),(4,121,54,97),(5,12,49,95),(6,125,50,101),(7,15,96,59),(9,13,92,57),(11,17,94,55),(14,98,58,122),(16,102,60,126),(18,100,56,124),(19,118,61,78),(20,128,62,105),(21,116,63,76),(22,132,64,103),(23,120,65,74),(24,130,66,107),(25,108,109,131),(26,73,110,119),(27,106,111,129),(28,77,112,117),(29,104,113,127),(30,75,114,115),(31,37,88,141),(32,47,89,136),(33,41,90,139),(34,45,85,134),(35,39,86,143),(36,43,87,138),(38,83,142,68),(40,81,144,72),(42,79,140,70),(44,82,133,67),(46,80,135,71),(48,84,137,69)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 6A | ··· | 6BH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 |
| kernel | C2×C62⋊5C4 | C62⋊5C4 | C22×C3⋊Dic3 | C22×C62 | C2×C62 | C23×C6 | C62 | C22×C6 | C22×C6 | C2×C6 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 4 | 16 | 12 | 32 |
Matrix representation of C2×C62⋊5C4 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 | 2 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,8,3],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,11,2] >;
C2×C62⋊5C4 in GAP, Magma, Sage, TeX
C_2\times C_6^2\rtimes_5C_4
% in TeX
G:=Group("C2xC6^2:5C4"); // GroupNames label
G:=SmallGroup(288,809);
// by ID
G=gap.SmallGroup(288,809);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^3,d*c*d^-1=c^-1>;
// generators/relations