direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊C8, C23⋊2C40, C24.3C20, (C22×C8)⋊1C10, (C22×C10)⋊6C8, C22⋊2(C2×C40), (C22×C40)⋊5C2, C4.69(D4×C10), (C2×C40)⋊42C22, (C2×C20).535D4, C20.474(C2×D4), C2.1(C22×C40), (C23×C20).8C2, (C23×C4).5C10, (C23×C10).11C4, (C22×C20).47C4, C10.54(C22×C8), (C22×C4).11C20, C23.28(C2×C20), C2.3(C10×M4(2)), (C2×C20).981C23, C10.81(C2×M4(2)), (C2×C10).48M4(2), C20.161(C22⋊C4), C22.9(C5×M4(2)), C22.19(C22×C20), (C22×C20).495C22, (C2×C8)⋊10(C2×C10), (C2×C10)⋊13(C2×C8), (C2×C4).59(C2×C20), (C2×C4).145(C5×D4), C2.3(C10×C22⋊C4), C4.31(C5×C22⋊C4), (C2×C20).461(C2×C4), C10.138(C2×C22⋊C4), C22.32(C5×C22⋊C4), (C2×C4).149(C22×C10), (C2×C10).332(C22×C4), (C22×C10).182(C2×C4), (C22×C4).135(C2×C10), (C2×C10).198(C22⋊C4), SmallGroup(320,907)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22⋊C8
G = < a,b,c,d | a10=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 290 in 202 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C23×C4, C40, C2×C20, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C2×C22⋊C8, C2×C40, C2×C40, C22×C20, C22×C20, C22×C20, C23×C10, C5×C22⋊C8, C22×C40, C23×C20, C10×C22⋊C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C23, C10, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C20, C2×C10, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C40, C2×C20, C5×D4, C22×C10, C2×C22⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C22×C20, D4×C10, C5×C22⋊C8, C10×C22⋊C4, C22×C40, C10×M4(2), C10×C22⋊C8
►On 160 points
Generators in S160
(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)
(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 41)(19 42)(20 43)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 31)(28 32)(29 33)(30 34)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 151)(60 152)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 41)(19 42)(20 43)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 31)(28 32)(29 33)(30 34)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 151)(60 152)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)(81 149)(82 150)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 131)(128 132)(129 133)(130 134)
(1 72 140 35 147 49 94 55)(2 73 131 36 148 50 95 56)(3 74 132 37 149 41 96 57)(4 75 133 38 150 42 97 58)(5 76 134 39 141 43 98 59)(6 77 135 40 142 44 99 60)(7 78 136 31 143 45 100 51)(8 79 137 32 144 46 91 52)(9 80 138 33 145 47 92 53)(10 71 139 34 146 48 93 54)(11 101 152 117 65 121 26 84)(12 102 153 118 66 122 27 85)(13 103 154 119 67 123 28 86)(14 104 155 120 68 124 29 87)(15 105 156 111 69 125 30 88)(16 106 157 112 70 126 21 89)(17 107 158 113 61 127 22 90)(18 108 159 114 62 128 23 81)(19 109 160 115 63 129 24 82)(20 110 151 116 64 130 25 83)
G:=sub<Sym(160)| (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), (11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,151)(60,152)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,151)(60,152)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72)(81,149)(82,150)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,131)(128,132)(129,133)(130,134), (1,72,140,35,147,49,94,55)(2,73,131,36,148,50,95,56)(3,74,132,37,149,41,96,57)(4,75,133,38,150,42,97,58)(5,76,134,39,141,43,98,59)(6,77,135,40,142,44,99,60)(7,78,136,31,143,45,100,51)(8,79,137,32,144,46,91,52)(9,80,138,33,145,47,92,53)(10,71,139,34,146,48,93,54)(11,101,152,117,65,121,26,84)(12,102,153,118,66,122,27,85)(13,103,154,119,67,123,28,86)(14,104,155,120,68,124,29,87)(15,105,156,111,69,125,30,88)(16,106,157,112,70,126,21,89)(17,107,158,113,61,127,22,90)(18,108,159,114,62,128,23,81)(19,109,160,115,63,129,24,82)(20,110,151,116,64,130,25,83)>;
G:=Group( (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), (11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,151)(60,152)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,41)(19,42)(20,43)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,159)(58,160)(59,151)(60,152)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72)(81,149)(82,150)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,131)(128,132)(129,133)(130,134), (1,72,140,35,147,49,94,55)(2,73,131,36,148,50,95,56)(3,74,132,37,149,41,96,57)(4,75,133,38,150,42,97,58)(5,76,134,39,141,43,98,59)(6,77,135,40,142,44,99,60)(7,78,136,31,143,45,100,51)(8,79,137,32,144,46,91,52)(9,80,138,33,145,47,92,53)(10,71,139,34,146,48,93,54)(11,101,152,117,65,121,26,84)(12,102,153,118,66,122,27,85)(13,103,154,119,67,123,28,86)(14,104,155,120,68,124,29,87)(15,105,156,111,69,125,30,88)(16,106,157,112,70,126,21,89)(17,107,158,113,61,127,22,90)(18,108,159,114,62,128,23,81)(19,109,160,115,63,129,24,82)(20,110,151,116,64,130,25,83) );
G=PermutationGroup([[(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)], [(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,41),(19,42),(20,43),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,31),(28,32),(29,33),(30,34),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,151),(60,152),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,71),(70,72)], [(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,111),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,41),(19,42),(20,43),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,31),(28,32),(29,33),(30,34),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,159),(58,160),(59,151),(60,152),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,71),(70,72),(81,149),(82,150),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,101),(100,102),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,131),(128,132),(129,133),(130,134)], [(1,72,140,35,147,49,94,55),(2,73,131,36,148,50,95,56),(3,74,132,37,149,41,96,57),(4,75,133,38,150,42,97,58),(5,76,134,39,141,43,98,59),(6,77,135,40,142,44,99,60),(7,78,136,31,143,45,100,51),(8,79,137,32,144,46,91,52),(9,80,138,33,145,47,92,53),(10,71,139,34,146,48,93,54),(11,101,152,117,65,121,26,84),(12,102,153,118,66,122,27,85),(13,103,154,119,67,123,28,86),(14,104,155,120,68,124,29,87),(15,105,156,111,69,125,30,88),(16,106,157,112,70,126,21,89),(17,107,158,113,61,127,22,90),(18,108,159,114,62,128,23,81),(19,109,160,115,63,129,24,82),(20,110,151,116,64,130,25,83)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20AV | 40A | ··· | 40BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | C5×D4 | C5×M4(2) |
| kernel | C10×C22⋊C8 | C5×C22⋊C8 | C22×C40 | C23×C20 | C22×C20 | C23×C10 | C2×C22⋊C8 | C22×C10 | C22⋊C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 16 | 16 | 8 | 4 | 24 | 8 | 64 | 4 | 4 | 16 | 16 |
Matrix representation of C10×C22⋊C8 ►in GL4(𝔽41) generated by
| 25 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 27 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 27 | 39 |
| 0 | 0 | 37 | 14 |
G:=sub<GL(4,GF(41))| [25,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,27,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,3,0,0,0,0,27,37,0,0,39,14] >;
C10×C22⋊C8 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes C_8 % in TeX
G:=Group("C10xC2^2:C8"); // GroupNames label
G:=SmallGroup(320,907);
// by ID
G=gap.SmallGroup(320,907);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations