direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊6D4, C40⋊28D4, C20⋊12M4(2), C8⋊6(C5×D4), C4⋊C8⋊16C10, (C4×C40)⋊31C2, (C4×C8)⋊15C10, C4⋊C4.8C20, (C2×D4).9C20, (C4×D4).3C10, C4.81(D4×C10), C2.11(D4×C20), C4⋊1(C5×M4(2)), C22⋊C8⋊14C10, (D4×C20).18C2, (D4×C10).34C4, C20.486(C2×D4), C10.143(C4×D4), C22⋊C4.5C20, C10.74(C8○D4), C42.69(C2×C10), C23.12(C2×C20), C20.355(C4○D4), (C2×M4(2))⋊15C10, (C10×M4(2))⋊33C2, (C4×C20).354C22, (C2×C20).992C23, (C2×C40).328C22, C2.10(C10×M4(2)), C10.88(C2×M4(2)), C22.48(C22×C20), (C22×C20).418C22, (C5×C4⋊C8)⋊35C2, C2.8(C5×C8○D4), (C5×C4⋊C4).33C4, C4.53(C5×C4○D4), (C5×C22⋊C8)⋊31C2, (C2×C8).53(C2×C10), (C2×C4).29(C2×C20), (C2×C20).386(C2×C4), (C5×C22⋊C4).20C4, (C22×C4).36(C2×C10), (C22×C10).92(C2×C4), (C2×C4).160(C22×C10), (C2×C10).343(C22×C4), SmallGroup(320,937)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊6D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C20, C20, C20, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C40, C40, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C8⋊6D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×M4(2), C22×C20, D4×C10, C4×C40, C5×C22⋊C8, C5×C4⋊C8, D4×C20, C10×M4(2), C5×C8⋊6D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, M4(2), C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×M4(2), C8○D4, C2×C20, C5×D4, C22×C10, C8⋊6D4, C5×M4(2), C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×M4(2), C5×C8○D4, C5×C8⋊6D4
►On 160 points
Generators in S160
(1 55 13 63 23)(2 56 14 64 24)(3 49 15 57 17)(4 50 16 58 18)(5 51 9 59 19)(6 52 10 60 20)(7 53 11 61 21)(8 54 12 62 22)(25 88 73 33 65)(26 81 74 34 66)(27 82 75 35 67)(28 83 76 36 68)(29 84 77 37 69)(30 85 78 38 70)(31 86 79 39 71)(32 87 80 40 72)(41 157 116 149 108)(42 158 117 150 109)(43 159 118 151 110)(44 160 119 152 111)(45 153 120 145 112)(46 154 113 146 105)(47 155 114 147 106)(48 156 115 148 107)(89 125 138 97 130)(90 126 139 98 131)(91 127 140 99 132)(92 128 141 100 133)(93 121 142 101 134)(94 122 143 102 135)(95 123 144 103 136)(96 124 137 104 129)
(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)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 105 27 89)(2 106 28 90)(3 107 29 91)(4 108 30 92)(5 109 31 93)(6 110 32 94)(7 111 25 95)(8 112 26 96)(9 158 79 142)(10 159 80 143)(11 160 73 144)(12 153 74 137)(13 154 75 138)(14 155 76 139)(15 156 77 140)(16 157 78 141)(17 148 69 132)(18 149 70 133)(19 150 71 134)(20 151 72 135)(21 152 65 136)(22 145 66 129)(23 146 67 130)(24 147 68 131)(33 103 61 119)(34 104 62 120)(35 97 63 113)(36 98 64 114)(37 99 57 115)(38 100 58 116)(39 101 59 117)(40 102 60 118)(41 85 128 50)(42 86 121 51)(43 87 122 52)(44 88 123 53)(45 81 124 54)(46 82 125 55)(47 83 126 56)(48 84 127 49)
(1 89)(2 94)(3 91)(4 96)(5 93)(6 90)(7 95)(8 92)(9 142)(10 139)(11 144)(12 141)(13 138)(14 143)(15 140)(16 137)(17 132)(18 129)(19 134)(20 131)(21 136)(22 133)(23 130)(24 135)(25 111)(26 108)(27 105)(28 110)(29 107)(30 112)(31 109)(32 106)(33 119)(34 116)(35 113)(36 118)(37 115)(38 120)(39 117)(40 114)(41 81)(42 86)(43 83)(44 88)(45 85)(46 82)(47 87)(48 84)(49 127)(50 124)(51 121)(52 126)(53 123)(54 128)(55 125)(56 122)(57 99)(58 104)(59 101)(60 98)(61 103)(62 100)(63 97)(64 102)(65 152)(66 149)(67 146)(68 151)(69 148)(70 145)(71 150)(72 147)(73 160)(74 157)(75 154)(76 159)(77 156)(78 153)(79 158)(80 155)
G:=sub<Sym(160)| (1,55,13,63,23)(2,56,14,64,24)(3,49,15,57,17)(4,50,16,58,18)(5,51,9,59,19)(6,52,10,60,20)(7,53,11,61,21)(8,54,12,62,22)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,157,116,149,108)(42,158,117,150,109)(43,159,118,151,110)(44,160,119,152,111)(45,153,120,145,112)(46,154,113,146,105)(47,155,114,147,106)(48,156,115,148,107)(89,125,138,97,130)(90,126,139,98,131)(91,127,140,99,132)(92,128,141,100,133)(93,121,142,101,134)(94,122,143,102,135)(95,123,144,103,136)(96,124,137,104,129), (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,105,27,89)(2,106,28,90)(3,107,29,91)(4,108,30,92)(5,109,31,93)(6,110,32,94)(7,111,25,95)(8,112,26,96)(9,158,79,142)(10,159,80,143)(11,160,73,144)(12,153,74,137)(13,154,75,138)(14,155,76,139)(15,156,77,140)(16,157,78,141)(17,148,69,132)(18,149,70,133)(19,150,71,134)(20,151,72,135)(21,152,65,136)(22,145,66,129)(23,146,67,130)(24,147,68,131)(33,103,61,119)(34,104,62,120)(35,97,63,113)(36,98,64,114)(37,99,57,115)(38,100,58,116)(39,101,59,117)(40,102,60,118)(41,85,128,50)(42,86,121,51)(43,87,122,52)(44,88,123,53)(45,81,124,54)(46,82,125,55)(47,83,126,56)(48,84,127,49), (1,89)(2,94)(3,91)(4,96)(5,93)(6,90)(7,95)(8,92)(9,142)(10,139)(11,144)(12,141)(13,138)(14,143)(15,140)(16,137)(17,132)(18,129)(19,134)(20,131)(21,136)(22,133)(23,130)(24,135)(25,111)(26,108)(27,105)(28,110)(29,107)(30,112)(31,109)(32,106)(33,119)(34,116)(35,113)(36,118)(37,115)(38,120)(39,117)(40,114)(41,81)(42,86)(43,83)(44,88)(45,85)(46,82)(47,87)(48,84)(49,127)(50,124)(51,121)(52,126)(53,123)(54,128)(55,125)(56,122)(57,99)(58,104)(59,101)(60,98)(61,103)(62,100)(63,97)(64,102)(65,152)(66,149)(67,146)(68,151)(69,148)(70,145)(71,150)(72,147)(73,160)(74,157)(75,154)(76,159)(77,156)(78,153)(79,158)(80,155)>;
G:=Group( (1,55,13,63,23)(2,56,14,64,24)(3,49,15,57,17)(4,50,16,58,18)(5,51,9,59,19)(6,52,10,60,20)(7,53,11,61,21)(8,54,12,62,22)(25,88,73,33,65)(26,81,74,34,66)(27,82,75,35,67)(28,83,76,36,68)(29,84,77,37,69)(30,85,78,38,70)(31,86,79,39,71)(32,87,80,40,72)(41,157,116,149,108)(42,158,117,150,109)(43,159,118,151,110)(44,160,119,152,111)(45,153,120,145,112)(46,154,113,146,105)(47,155,114,147,106)(48,156,115,148,107)(89,125,138,97,130)(90,126,139,98,131)(91,127,140,99,132)(92,128,141,100,133)(93,121,142,101,134)(94,122,143,102,135)(95,123,144,103,136)(96,124,137,104,129), (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,105,27,89)(2,106,28,90)(3,107,29,91)(4,108,30,92)(5,109,31,93)(6,110,32,94)(7,111,25,95)(8,112,26,96)(9,158,79,142)(10,159,80,143)(11,160,73,144)(12,153,74,137)(13,154,75,138)(14,155,76,139)(15,156,77,140)(16,157,78,141)(17,148,69,132)(18,149,70,133)(19,150,71,134)(20,151,72,135)(21,152,65,136)(22,145,66,129)(23,146,67,130)(24,147,68,131)(33,103,61,119)(34,104,62,120)(35,97,63,113)(36,98,64,114)(37,99,57,115)(38,100,58,116)(39,101,59,117)(40,102,60,118)(41,85,128,50)(42,86,121,51)(43,87,122,52)(44,88,123,53)(45,81,124,54)(46,82,125,55)(47,83,126,56)(48,84,127,49), (1,89)(2,94)(3,91)(4,96)(5,93)(6,90)(7,95)(8,92)(9,142)(10,139)(11,144)(12,141)(13,138)(14,143)(15,140)(16,137)(17,132)(18,129)(19,134)(20,131)(21,136)(22,133)(23,130)(24,135)(25,111)(26,108)(27,105)(28,110)(29,107)(30,112)(31,109)(32,106)(33,119)(34,116)(35,113)(36,118)(37,115)(38,120)(39,117)(40,114)(41,81)(42,86)(43,83)(44,88)(45,85)(46,82)(47,87)(48,84)(49,127)(50,124)(51,121)(52,126)(53,123)(54,128)(55,125)(56,122)(57,99)(58,104)(59,101)(60,98)(61,103)(62,100)(63,97)(64,102)(65,152)(66,149)(67,146)(68,151)(69,148)(70,145)(71,150)(72,147)(73,160)(74,157)(75,154)(76,159)(77,156)(78,153)(79,158)(80,155) );
G=PermutationGroup([[(1,55,13,63,23),(2,56,14,64,24),(3,49,15,57,17),(4,50,16,58,18),(5,51,9,59,19),(6,52,10,60,20),(7,53,11,61,21),(8,54,12,62,22),(25,88,73,33,65),(26,81,74,34,66),(27,82,75,35,67),(28,83,76,36,68),(29,84,77,37,69),(30,85,78,38,70),(31,86,79,39,71),(32,87,80,40,72),(41,157,116,149,108),(42,158,117,150,109),(43,159,118,151,110),(44,160,119,152,111),(45,153,120,145,112),(46,154,113,146,105),(47,155,114,147,106),(48,156,115,148,107),(89,125,138,97,130),(90,126,139,98,131),(91,127,140,99,132),(92,128,141,100,133),(93,121,142,101,134),(94,122,143,102,135),(95,123,144,103,136),(96,124,137,104,129)], [(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),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,105,27,89),(2,106,28,90),(3,107,29,91),(4,108,30,92),(5,109,31,93),(6,110,32,94),(7,111,25,95),(8,112,26,96),(9,158,79,142),(10,159,80,143),(11,160,73,144),(12,153,74,137),(13,154,75,138),(14,155,76,139),(15,156,77,140),(16,157,78,141),(17,148,69,132),(18,149,70,133),(19,150,71,134),(20,151,72,135),(21,152,65,136),(22,145,66,129),(23,146,67,130),(24,147,68,131),(33,103,61,119),(34,104,62,120),(35,97,63,113),(36,98,64,114),(37,99,57,115),(38,100,58,116),(39,101,59,117),(40,102,60,118),(41,85,128,50),(42,86,121,51),(43,87,122,52),(44,88,123,53),(45,81,124,54),(46,82,125,55),(47,83,126,56),(48,84,127,49)], [(1,89),(2,94),(3,91),(4,96),(5,93),(6,90),(7,95),(8,92),(9,142),(10,139),(11,144),(12,141),(13,138),(14,143),(15,140),(16,137),(17,132),(18,129),(19,134),(20,131),(21,136),(22,133),(23,130),(24,135),(25,111),(26,108),(27,105),(28,110),(29,107),(30,112),(31,109),(32,106),(33,119),(34,116),(35,113),(36,118),(37,115),(38,120),(39,117),(40,114),(41,81),(42,86),(43,83),(44,88),(45,85),(46,82),(47,87),(48,84),(49,127),(50,124),(51,121),(52,126),(53,123),(54,128),(55,125),(56,122),(57,99),(58,104),(59,101),(60,98),(61,103),(62,100),(63,97),(64,102),(65,152),(66,149),(67,146),(68,151),(69,148),(70,145),(71,150),(72,147),(73,160),(74,157),(75,154),(76,159),(77,156),(78,153),(79,158),(80,155)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | M4(2) | C4○D4 | C8○D4 | C5×D4 | C5×M4(2) | C5×C4○D4 | C5×C8○D4 |
| kernel | C5×C8⋊6D4 | C4×C40 | C5×C22⋊C8 | C5×C4⋊C8 | D4×C20 | C10×M4(2) | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C8⋊6D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C40 | C20 | C20 | C10 | C8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 8 | 8 | 2 | 4 | 2 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C5×C8⋊6D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 39 | 0 | 0 |
| 21 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 9 | 0 |
| 34 | 6 | 0 | 0 |
| 19 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 6 | 0 | 0 |
| 33 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,21,0,0,39,25,0,0,0,0,0,9,0,0,1,0],[34,19,0,0,6,7,0,0,0,0,1,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;
C5×C8⋊6D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_6D_4
% in TeX
G:=Group("C5xC8:6D4"); // GroupNames label
G:=SmallGroup(320,937);
// by ID
G=gap.SmallGroup(320,937);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1731,436,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations