direct product, metabelian, supersoluble, monomial
Aliases: C3×C32⋊5D8, C33⋊11D8, C32⋊7D24, C24⋊1(C3×S3), (C3×C24)⋊9C6, C3⋊1(C3×D24), C24⋊3(C3⋊S3), (C3×C24)⋊10S3, C12⋊S3⋊7C6, C32⋊8(C3×D8), C12.81(S3×C6), (C32×C24)⋊3C2, C6.20(C3×D12), (C3×C6).62D12, (C3×C12).208D6, (C32×C6).58D4, C6.23(C12⋊S3), (C32×C12).83C22, C8⋊1(C3×C3⋊S3), C4.9(C6×C3⋊S3), C12.85(C2×C3⋊S3), (C3×C6).50(C3×D4), C2.4(C3×C12⋊S3), (C3×C12).73(C2×C6), (C3×C12⋊S3)⋊16C2, SmallGroup(432,483)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊5D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 772 in 164 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, C3×D8, C3×C3⋊S3, C32×C6, C3×C24, C3×C24, C3×C24, C3×D12, C12⋊S3, C32×C12, C6×C3⋊S3, C3×D24, C32⋊5D8, C32×C24, C3×C12⋊S3, C3×C32⋊5D8
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, D24, C3×D8, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×D24, C32⋊5D8, C3×C12⋊S3, C3×C32⋊5D8
►On 144 points
Generators in S144
(1 88 104)(2 81 97)(3 82 98)(4 83 99)(5 84 100)(6 85 101)(7 86 102)(8 87 103)(9 22 61)(10 23 62)(11 24 63)(12 17 64)(13 18 57)(14 19 58)(15 20 59)(16 21 60)(25 38 44)(26 39 45)(27 40 46)(28 33 47)(29 34 48)(30 35 41)(31 36 42)(32 37 43)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 129 77)(66 130 78)(67 131 79)(68 132 80)(69 133 73)(70 134 74)(71 135 75)(72 136 76)(105 124 113)(106 125 114)(107 126 115)(108 127 116)(109 128 117)(110 121 118)(111 122 119)(112 123 120)
(1 78 118)(2 79 119)(3 80 120)(4 73 113)(5 74 114)(6 75 115)(7 76 116)(8 77 117)(9 47 142)(10 48 143)(11 41 144)(12 42 137)(13 43 138)(14 44 139)(15 45 140)(16 46 141)(17 31 95)(18 32 96)(19 25 89)(20 26 90)(21 27 91)(22 28 92)(23 29 93)(24 30 94)(33 54 61)(34 55 62)(35 56 63)(36 49 64)(37 50 57)(38 51 58)(39 52 59)(40 53 60)(65 109 87)(66 110 88)(67 111 81)(68 112 82)(69 105 83)(70 106 84)(71 107 85)(72 108 86)(97 131 122)(98 132 123)(99 133 124)(100 134 125)(101 135 126)(102 136 127)(103 129 128)(104 130 121)
(1 104 88)(2 97 81)(3 98 82)(4 99 83)(5 100 84)(6 101 85)(7 102 86)(8 103 87)(9 22 61)(10 23 62)(11 24 63)(12 17 64)(13 18 57)(14 19 58)(15 20 59)(16 21 60)(25 38 44)(26 39 45)(27 40 46)(28 33 47)(29 34 48)(30 35 41)(31 36 42)(32 37 43)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 77 129)(66 78 130)(67 79 131)(68 80 132)(69 73 133)(70 74 134)(71 75 135)(72 76 136)(105 113 124)(106 114 125)(107 115 126)(108 116 127)(109 117 128)(110 118 121)(111 119 122)(112 120 123)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)(17 82)(18 81)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 110)(26 109)(27 108)(28 107)(29 106)(30 105)(31 112)(32 111)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 128)(40 127)(41 113)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 132)(50 131)(51 130)(52 129)(53 136)(54 135)(55 134)(56 133)(57 97)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 90)(66 89)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 144)(74 143)(75 142)(76 141)(77 140)(78 139)(79 138)(80 137)
G:=sub<Sym(144)| (1,88,104)(2,81,97)(3,82,98)(4,83,99)(5,84,100)(6,85,101)(7,86,102)(8,87,103)(9,22,61)(10,23,62)(11,24,63)(12,17,64)(13,18,57)(14,19,58)(15,20,59)(16,21,60)(25,38,44)(26,39,45)(27,40,46)(28,33,47)(29,34,48)(30,35,41)(31,36,42)(32,37,43)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,129,77)(66,130,78)(67,131,79)(68,132,80)(69,133,73)(70,134,74)(71,135,75)(72,136,76)(105,124,113)(106,125,114)(107,126,115)(108,127,116)(109,128,117)(110,121,118)(111,122,119)(112,123,120), (1,78,118)(2,79,119)(3,80,120)(4,73,113)(5,74,114)(6,75,115)(7,76,116)(8,77,117)(9,47,142)(10,48,143)(11,41,144)(12,42,137)(13,43,138)(14,44,139)(15,45,140)(16,46,141)(17,31,95)(18,32,96)(19,25,89)(20,26,90)(21,27,91)(22,28,92)(23,29,93)(24,30,94)(33,54,61)(34,55,62)(35,56,63)(36,49,64)(37,50,57)(38,51,58)(39,52,59)(40,53,60)(65,109,87)(66,110,88)(67,111,81)(68,112,82)(69,105,83)(70,106,84)(71,107,85)(72,108,86)(97,131,122)(98,132,123)(99,133,124)(100,134,125)(101,135,126)(102,136,127)(103,129,128)(104,130,121), (1,104,88)(2,97,81)(3,98,82)(4,99,83)(5,100,84)(6,101,85)(7,102,86)(8,103,87)(9,22,61)(10,23,62)(11,24,63)(12,17,64)(13,18,57)(14,19,58)(15,20,59)(16,21,60)(25,38,44)(26,39,45)(27,40,46)(28,33,47)(29,34,48)(30,35,41)(31,36,42)(32,37,43)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,77,129)(66,78,130)(67,79,131)(68,80,132)(69,73,133)(70,74,134)(71,75,135)(72,76,136)(105,113,124)(106,114,125)(107,115,126)(108,116,127)(109,117,128)(110,118,121)(111,119,122)(112,120,123), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15)(17,82)(18,81)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,112)(32,111)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,128)(40,127)(41,113)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,132)(50,131)(51,130)(52,129)(53,136)(54,135)(55,134)(56,133)(57,97)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,90)(66,89)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)>;
G:=Group( (1,88,104)(2,81,97)(3,82,98)(4,83,99)(5,84,100)(6,85,101)(7,86,102)(8,87,103)(9,22,61)(10,23,62)(11,24,63)(12,17,64)(13,18,57)(14,19,58)(15,20,59)(16,21,60)(25,38,44)(26,39,45)(27,40,46)(28,33,47)(29,34,48)(30,35,41)(31,36,42)(32,37,43)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,129,77)(66,130,78)(67,131,79)(68,132,80)(69,133,73)(70,134,74)(71,135,75)(72,136,76)(105,124,113)(106,125,114)(107,126,115)(108,127,116)(109,128,117)(110,121,118)(111,122,119)(112,123,120), (1,78,118)(2,79,119)(3,80,120)(4,73,113)(5,74,114)(6,75,115)(7,76,116)(8,77,117)(9,47,142)(10,48,143)(11,41,144)(12,42,137)(13,43,138)(14,44,139)(15,45,140)(16,46,141)(17,31,95)(18,32,96)(19,25,89)(20,26,90)(21,27,91)(22,28,92)(23,29,93)(24,30,94)(33,54,61)(34,55,62)(35,56,63)(36,49,64)(37,50,57)(38,51,58)(39,52,59)(40,53,60)(65,109,87)(66,110,88)(67,111,81)(68,112,82)(69,105,83)(70,106,84)(71,107,85)(72,108,86)(97,131,122)(98,132,123)(99,133,124)(100,134,125)(101,135,126)(102,136,127)(103,129,128)(104,130,121), (1,104,88)(2,97,81)(3,98,82)(4,99,83)(5,100,84)(6,101,85)(7,102,86)(8,103,87)(9,22,61)(10,23,62)(11,24,63)(12,17,64)(13,18,57)(14,19,58)(15,20,59)(16,21,60)(25,38,44)(26,39,45)(27,40,46)(28,33,47)(29,34,48)(30,35,41)(31,36,42)(32,37,43)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,77,129)(66,78,130)(67,79,131)(68,80,132)(69,73,133)(70,74,134)(71,75,135)(72,76,136)(105,113,124)(106,114,125)(107,115,126)(108,116,127)(109,117,128)(110,118,121)(111,119,122)(112,120,123), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15)(17,82)(18,81)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,112)(32,111)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,128)(40,127)(41,113)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,132)(50,131)(51,130)(52,129)(53,136)(54,135)(55,134)(56,133)(57,97)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,90)(66,89)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137) );
G=PermutationGroup([[(1,88,104),(2,81,97),(3,82,98),(4,83,99),(5,84,100),(6,85,101),(7,86,102),(8,87,103),(9,22,61),(10,23,62),(11,24,63),(12,17,64),(13,18,57),(14,19,58),(15,20,59),(16,21,60),(25,38,44),(26,39,45),(27,40,46),(28,33,47),(29,34,48),(30,35,41),(31,36,42),(32,37,43),(49,137,95),(50,138,96),(51,139,89),(52,140,90),(53,141,91),(54,142,92),(55,143,93),(56,144,94),(65,129,77),(66,130,78),(67,131,79),(68,132,80),(69,133,73),(70,134,74),(71,135,75),(72,136,76),(105,124,113),(106,125,114),(107,126,115),(108,127,116),(109,128,117),(110,121,118),(111,122,119),(112,123,120)], [(1,78,118),(2,79,119),(3,80,120),(4,73,113),(5,74,114),(6,75,115),(7,76,116),(8,77,117),(9,47,142),(10,48,143),(11,41,144),(12,42,137),(13,43,138),(14,44,139),(15,45,140),(16,46,141),(17,31,95),(18,32,96),(19,25,89),(20,26,90),(21,27,91),(22,28,92),(23,29,93),(24,30,94),(33,54,61),(34,55,62),(35,56,63),(36,49,64),(37,50,57),(38,51,58),(39,52,59),(40,53,60),(65,109,87),(66,110,88),(67,111,81),(68,112,82),(69,105,83),(70,106,84),(71,107,85),(72,108,86),(97,131,122),(98,132,123),(99,133,124),(100,134,125),(101,135,126),(102,136,127),(103,129,128),(104,130,121)], [(1,104,88),(2,97,81),(3,98,82),(4,99,83),(5,100,84),(6,101,85),(7,102,86),(8,103,87),(9,22,61),(10,23,62),(11,24,63),(12,17,64),(13,18,57),(14,19,58),(15,20,59),(16,21,60),(25,38,44),(26,39,45),(27,40,46),(28,33,47),(29,34,48),(30,35,41),(31,36,42),(32,37,43),(49,137,95),(50,138,96),(51,139,89),(52,140,90),(53,141,91),(54,142,92),(55,143,93),(56,144,94),(65,77,129),(66,78,130),(67,79,131),(68,80,132),(69,73,133),(70,74,134),(71,75,135),(72,76,136),(105,113,124),(106,114,125),(107,115,126),(108,116,127),(109,117,128),(110,118,121),(111,119,122),(112,120,123)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,16),(8,15),(17,82),(18,81),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,110),(26,109),(27,108),(28,107),(29,106),(30,105),(31,112),(32,111),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,128),(40,127),(41,113),(42,120),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,132),(50,131),(51,130),(52,129),(53,136),(54,135),(55,134),(56,133),(57,97),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,90),(66,89),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,144),(74,143),(75,142),(76,141),(77,140),(78,139),(79,138),(80,137)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4 | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 6Q | 6R | 8A | 8B | 12A | ··· | 12Z | 24A | ··· | 24AZ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 36 | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | D12 | C3×D4 | S3×C6 | D24 | C3×D8 | C3×D12 | C3×D24 |
| kernel | C3×C32⋊5D8 | C32×C24 | C3×C12⋊S3 | C32⋊5D8 | C3×C24 | C12⋊S3 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 2 | 8 | 8 | 2 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C3×C32⋊5D8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 51 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 63 | 0 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,64,0,0,0,0,72,72,0,0,1,0],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[51,0,0,0,0,63,0,0,0,0,72,0,0,0,0,72],[0,51,0,0,63,0,0,0,0,0,72,0,0,0,1,1] >;
C3×C32⋊5D8 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_5D_8
% in TeX
G:=Group("C3xC3^2:5D8"); // GroupNames label
G:=SmallGroup(432,483);
// by ID
G=gap.SmallGroup(432,483);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations