direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊7D4, C40⋊33D4, C8⋊7(C5×D4), (C2×C10)⋊4D8, (C2×D8)⋊4C10, C2.D8⋊2C10, C2.6(C10×D8), C22⋊1(C5×D8), (C10×D8)⋊18C2, C4⋊D4⋊3C10, (C22×C8)⋊6C10, C4.58(D4×C10), C10.78(C2×D8), D4⋊C4⋊2C10, (C22×C40)⋊20C2, (C2×C20).364D4, C20.465(C2×D4), C23.26(C5×D4), C22.90(D4×C10), C10.124(C4○D8), C20.263(C4○D4), (C2×C20).925C23, (C2×C40).364C22, (C22×C10).130D4, C10.149(C4⋊D4), (D4×C10).190C22, (C22×C20).592C22, C4.8(C5×C4○D4), C4⋊C4.6(C2×C10), (C5×C2.D8)⋊17C2, C2.11(C5×C4○D8), (C2×C4).54(C5×D4), (C5×D4⋊C4)⋊2C2, (C5×C4⋊D4)⋊30C2, (C2×C8).77(C2×C10), C2.18(C5×C4⋊D4), (C2×D4).13(C2×C10), (C2×C10).646(C2×D4), (C5×C4⋊C4).228C22, (C2×C4).100(C22×C10), (C22×C4).121(C2×C10), SmallGroup(320,967)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊7D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 282 in 134 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C40, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C8⋊7D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×D8, C22×C20, D4×C10, D4×C10, C5×D4⋊C4, C5×C2.D8, C5×C4⋊D4, C22×C40, C10×D8, C5×C8⋊7D4
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×D8, C4○D8, C5×D4, C22×C10, C8⋊7D4, C5×D8, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×D8, C5×C4○D8, C5×C8⋊7D4
►On 160 points
Generators in S160
(1 139 57 131 49)(2 140 58 132 50)(3 141 59 133 51)(4 142 60 134 52)(5 143 61 135 53)(6 144 62 136 54)(7 137 63 129 55)(8 138 64 130 56)(9 109 99 94 19)(10 110 100 95 20)(11 111 101 96 21)(12 112 102 89 22)(13 105 103 90 23)(14 106 104 91 24)(15 107 97 92 17)(16 108 98 93 18)(25 124 115 48 38)(26 125 116 41 39)(27 126 117 42 40)(28 127 118 43 33)(29 128 119 44 34)(30 121 120 45 35)(31 122 113 46 36)(32 123 114 47 37)(65 82 155 73 147)(66 83 156 74 148)(67 84 157 75 149)(68 85 158 76 150)(69 86 159 77 151)(70 87 160 78 152)(71 88 153 79 145)(72 81 154 80 146)
(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 31 13 72)(2 30 14 71)(3 29 15 70)(4 28 16 69)(5 27 9 68)(6 26 10 67)(7 25 11 66)(8 32 12 65)(17 152 51 34)(18 151 52 33)(19 150 53 40)(20 149 54 39)(21 148 55 38)(22 147 56 37)(23 146 49 36)(24 145 50 35)(41 95 75 136)(42 94 76 135)(43 93 77 134)(44 92 78 133)(45 91 79 132)(46 90 80 131)(47 89 73 130)(48 96 74 129)(57 113 103 154)(58 120 104 153)(59 119 97 160)(60 118 98 159)(61 117 99 158)(62 116 100 157)(63 115 101 156)(64 114 102 155)(81 139 122 105)(82 138 123 112)(83 137 124 111)(84 144 125 110)(85 143 126 109)(86 142 127 108)(87 141 128 107)(88 140 121 106)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 56)(24 55)(25 30)(26 29)(27 28)(31 32)(33 40)(34 39)(35 38)(36 37)(41 44)(42 43)(45 48)(46 47)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 104)(64 103)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 82)(83 88)(84 87)(85 86)(89 131)(90 130)(91 129)(92 136)(93 135)(94 134)(95 133)(96 132)(105 138)(106 137)(107 144)(108 143)(109 142)(110 141)(111 140)(112 139)(113 114)(115 120)(116 119)(117 118)(121 124)(122 123)(125 128)(126 127)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)
G:=sub<Sym(160)| (1,139,57,131,49)(2,140,58,132,50)(3,141,59,133,51)(4,142,60,134,52)(5,143,61,135,53)(6,144,62,136,54)(7,137,63,129,55)(8,138,64,130,56)(9,109,99,94,19)(10,110,100,95,20)(11,111,101,96,21)(12,112,102,89,22)(13,105,103,90,23)(14,106,104,91,24)(15,107,97,92,17)(16,108,98,93,18)(25,124,115,48,38)(26,125,116,41,39)(27,126,117,42,40)(28,127,118,43,33)(29,128,119,44,34)(30,121,120,45,35)(31,122,113,46,36)(32,123,114,47,37)(65,82,155,73,147)(66,83,156,74,148)(67,84,157,75,149)(68,85,158,76,150)(69,86,159,77,151)(70,87,160,78,152)(71,88,153,79,145)(72,81,154,80,146), (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), (1,31,13,72)(2,30,14,71)(3,29,15,70)(4,28,16,69)(5,27,9,68)(6,26,10,67)(7,25,11,66)(8,32,12,65)(17,152,51,34)(18,151,52,33)(19,150,53,40)(20,149,54,39)(21,148,55,38)(22,147,56,37)(23,146,49,36)(24,145,50,35)(41,95,75,136)(42,94,76,135)(43,93,77,134)(44,92,78,133)(45,91,79,132)(46,90,80,131)(47,89,73,130)(48,96,74,129)(57,113,103,154)(58,120,104,153)(59,119,97,160)(60,118,98,159)(61,117,99,158)(62,116,100,157)(63,115,101,156)(64,114,102,155)(81,139,122,105)(82,138,123,112)(83,137,124,111)(84,144,125,110)(85,143,126,109)(86,142,127,108)(87,141,128,107)(88,140,121,106), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,56)(24,55)(25,30)(26,29)(27,28)(31,32)(33,40)(34,39)(35,38)(36,37)(41,44)(42,43)(45,48)(46,47)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,104)(64,103)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,131)(90,130)(91,129)(92,136)(93,135)(94,134)(95,133)(96,132)(105,138)(106,137)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,114)(115,120)(116,119)(117,118)(121,124)(122,123)(125,128)(126,127)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)>;
G:=Group( (1,139,57,131,49)(2,140,58,132,50)(3,141,59,133,51)(4,142,60,134,52)(5,143,61,135,53)(6,144,62,136,54)(7,137,63,129,55)(8,138,64,130,56)(9,109,99,94,19)(10,110,100,95,20)(11,111,101,96,21)(12,112,102,89,22)(13,105,103,90,23)(14,106,104,91,24)(15,107,97,92,17)(16,108,98,93,18)(25,124,115,48,38)(26,125,116,41,39)(27,126,117,42,40)(28,127,118,43,33)(29,128,119,44,34)(30,121,120,45,35)(31,122,113,46,36)(32,123,114,47,37)(65,82,155,73,147)(66,83,156,74,148)(67,84,157,75,149)(68,85,158,76,150)(69,86,159,77,151)(70,87,160,78,152)(71,88,153,79,145)(72,81,154,80,146), (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), (1,31,13,72)(2,30,14,71)(3,29,15,70)(4,28,16,69)(5,27,9,68)(6,26,10,67)(7,25,11,66)(8,32,12,65)(17,152,51,34)(18,151,52,33)(19,150,53,40)(20,149,54,39)(21,148,55,38)(22,147,56,37)(23,146,49,36)(24,145,50,35)(41,95,75,136)(42,94,76,135)(43,93,77,134)(44,92,78,133)(45,91,79,132)(46,90,80,131)(47,89,73,130)(48,96,74,129)(57,113,103,154)(58,120,104,153)(59,119,97,160)(60,118,98,159)(61,117,99,158)(62,116,100,157)(63,115,101,156)(64,114,102,155)(81,139,122,105)(82,138,123,112)(83,137,124,111)(84,144,125,110)(85,143,126,109)(86,142,127,108)(87,141,128,107)(88,140,121,106), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,56)(24,55)(25,30)(26,29)(27,28)(31,32)(33,40)(34,39)(35,38)(36,37)(41,44)(42,43)(45,48)(46,47)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,104)(64,103)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,131)(90,130)(91,129)(92,136)(93,135)(94,134)(95,133)(96,132)(105,138)(106,137)(107,144)(108,143)(109,142)(110,141)(111,140)(112,139)(113,114)(115,120)(116,119)(117,118)(121,124)(122,123)(125,128)(126,127)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159) );
G=PermutationGroup([[(1,139,57,131,49),(2,140,58,132,50),(3,141,59,133,51),(4,142,60,134,52),(5,143,61,135,53),(6,144,62,136,54),(7,137,63,129,55),(8,138,64,130,56),(9,109,99,94,19),(10,110,100,95,20),(11,111,101,96,21),(12,112,102,89,22),(13,105,103,90,23),(14,106,104,91,24),(15,107,97,92,17),(16,108,98,93,18),(25,124,115,48,38),(26,125,116,41,39),(27,126,117,42,40),(28,127,118,43,33),(29,128,119,44,34),(30,121,120,45,35),(31,122,113,46,36),(32,123,114,47,37),(65,82,155,73,147),(66,83,156,74,148),(67,84,157,75,149),(68,85,158,76,150),(69,86,159,77,151),(70,87,160,78,152),(71,88,153,79,145),(72,81,154,80,146)], [(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)], [(1,31,13,72),(2,30,14,71),(3,29,15,70),(4,28,16,69),(5,27,9,68),(6,26,10,67),(7,25,11,66),(8,32,12,65),(17,152,51,34),(18,151,52,33),(19,150,53,40),(20,149,54,39),(21,148,55,38),(22,147,56,37),(23,146,49,36),(24,145,50,35),(41,95,75,136),(42,94,76,135),(43,93,77,134),(44,92,78,133),(45,91,79,132),(46,90,80,131),(47,89,73,130),(48,96,74,129),(57,113,103,154),(58,120,104,153),(59,119,97,160),(60,118,98,159),(61,117,99,158),(62,116,100,157),(63,115,101,156),(64,114,102,155),(81,139,122,105),(82,138,123,112),(83,137,124,111),(84,144,125,110),(85,143,126,109),(86,142,127,108),(87,141,128,107),(88,140,121,106)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,56),(24,55),(25,30),(26,29),(27,28),(31,32),(33,40),(34,39),(35,38),(36,37),(41,44),(42,43),(45,48),(46,47),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,104),(64,103),(65,72),(66,71),(67,70),(68,69),(73,80),(74,79),(75,78),(76,77),(81,82),(83,88),(84,87),(85,86),(89,131),(90,130),(91,129),(92,136),(93,135),(94,134),(95,133),(96,132),(105,138),(106,137),(107,144),(108,143),(109,142),(110,141),(111,140),(112,139),(113,114),(115,120),(116,119),(117,118),(121,124),(122,123),(125,128),(126,127),(145,148),(146,147),(149,152),(150,151),(153,156),(154,155),(157,160),(158,159)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | D8 | C4○D8 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | C5×D8 | C5×C4○D8 |
| kernel | C5×C8⋊7D4 | C5×D4⋊C4 | C5×C2.D8 | C5×C4⋊D4 | C22×C40 | C10×D8 | C8⋊7D4 | D4⋊C4 | C2.D8 | C4⋊D4 | C22×C8 | C2×D8 | C40 | C2×C20 | C22×C10 | C20 | C2×C10 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C5×C8⋊7D4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 29 | 12 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 15 | 15 | 0 | 0 |
| 15 | 26 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 12 | 12 |
| 12 | 29 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 29 | 29 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[29,29,0,0,12,29,0,0,0,0,12,12,0,0,29,12],[15,15,0,0,15,26,0,0,0,0,29,12,0,0,12,12],[12,29,0,0,29,29,0,0,0,0,12,29,0,0,29,29] >;
C5×C8⋊7D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_7D_4
% in TeX
G:=Group("C5xC8:7D4"); // GroupNames label
G:=SmallGroup(320,967);
// by ID
G=gap.SmallGroup(320,967);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,7004,172]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations