metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊2C8, C5⋊C8⋊9D4, C5⋊2(C8×D4), C20⋊1(C2×C8), C4⋊1(D5⋊C8), C2.3(D4×F5), D10⋊2(C2×C8), C20⋊C8⋊3C2, C10.8(C4×D4), C4⋊C4.12F5, (C2×D20).11C4, C10.6(C22×C8), D10⋊C8⋊13C2, C2.2(Q8.F5), D10⋊C4.8C4, C10.19(C8○D4), Dic5.69(C2×D4), D20⋊8C4.16C2, Dic5.54(C4○D4), C22.37(C22×F5), (C4×Dic5).190C22, (C2×Dic5).329C23, (C4×C5⋊C8)⋊3C2, (C5×C4⋊C4).6C4, C2.8(C2×D5⋊C8), (C2×D5⋊C8)⋊10C2, (C2×C4).60(C2×F5), (C2×C20).42(C2×C4), (C2×C5⋊C8).26C22, (C2×C4×D5).289C22, (C2×C10).40(C22×C4), (C2×Dic5).55(C2×C4), (C22×D5).47(C2×C4), SmallGroup(320,1040)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 474 in 134 conjugacy classes, 56 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C8×D4, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, D5⋊C8 [×4], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C4×D5 [×2], C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8 [×2], D20⋊8C4, C2×D5⋊C8 [×2], D20⋊2C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C2×F5 [×3], C8×D4, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D4×F5, Q8.F5, D20⋊2C8
Generators and relations
G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a12b >
►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)
(1 143)(2 142)(3 141)(4 160)(5 159)(6 158)(7 157)(8 156)(9 155)(10 154)(11 153)(12 152)(13 151)(14 150)(15 149)(16 148)(17 147)(18 146)(19 145)(20 144)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)
(1 33 103 53 149 68 98 123)(2 30 112 46 150 65 87 136)(3 27 101 59 151 62 96 129)(4 24 110 52 152 79 85 122)(5 21 119 45 153 76 94 135)(6 38 108 58 154 73 83 128)(7 35 117 51 155 70 92 121)(8 32 106 44 156 67 81 134)(9 29 115 57 157 64 90 127)(10 26 104 50 158 61 99 140)(11 23 113 43 159 78 88 133)(12 40 102 56 160 75 97 126)(13 37 111 49 141 72 86 139)(14 34 120 42 142 69 95 132)(15 31 109 55 143 66 84 125)(16 28 118 48 144 63 93 138)(17 25 107 41 145 80 82 131)(18 22 116 54 146 77 91 124)(19 39 105 47 147 74 100 137)(20 36 114 60 148 71 89 130)
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), (1,143)(2,142)(3,141)(4,160)(5,159)(6,158)(7,157)(8,156)(9,155)(10,154)(11,153)(12,152)(13,151)(14,150)(15,149)(16,148)(17,147)(18,146)(19,145)(20,144)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115), (1,33,103,53,149,68,98,123)(2,30,112,46,150,65,87,136)(3,27,101,59,151,62,96,129)(4,24,110,52,152,79,85,122)(5,21,119,45,153,76,94,135)(6,38,108,58,154,73,83,128)(7,35,117,51,155,70,92,121)(8,32,106,44,156,67,81,134)(9,29,115,57,157,64,90,127)(10,26,104,50,158,61,99,140)(11,23,113,43,159,78,88,133)(12,40,102,56,160,75,97,126)(13,37,111,49,141,72,86,139)(14,34,120,42,142,69,95,132)(15,31,109,55,143,66,84,125)(16,28,118,48,144,63,93,138)(17,25,107,41,145,80,82,131)(18,22,116,54,146,77,91,124)(19,39,105,47,147,74,100,137)(20,36,114,60,148,71,89,130)>;
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), (1,143)(2,142)(3,141)(4,160)(5,159)(6,158)(7,157)(8,156)(9,155)(10,154)(11,153)(12,152)(13,151)(14,150)(15,149)(16,148)(17,147)(18,146)(19,145)(20,144)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115), (1,33,103,53,149,68,98,123)(2,30,112,46,150,65,87,136)(3,27,101,59,151,62,96,129)(4,24,110,52,152,79,85,122)(5,21,119,45,153,76,94,135)(6,38,108,58,154,73,83,128)(7,35,117,51,155,70,92,121)(8,32,106,44,156,67,81,134)(9,29,115,57,157,64,90,127)(10,26,104,50,158,61,99,140)(11,23,113,43,159,78,88,133)(12,40,102,56,160,75,97,126)(13,37,111,49,141,72,86,139)(14,34,120,42,142,69,95,132)(15,31,109,55,143,66,84,125)(16,28,118,48,144,63,93,138)(17,25,107,41,145,80,82,131)(18,22,116,54,146,77,91,124)(19,39,105,47,147,74,100,137)(20,36,114,60,148,71,89,130) );
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)], [(1,143),(2,142),(3,141),(4,160),(5,159),(6,158),(7,157),(8,156),(9,155),(10,154),(11,153),(12,152),(13,151),(14,150),(15,149),(16,148),(17,147),(18,146),(19,145),(20,144),(21,74),(22,73),(23,72),(24,71),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,80),(36,79),(37,78),(38,77),(39,76),(40,75),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,133),(58,132),(59,131),(60,130),(81,114),(82,113),(83,112),(84,111),(85,110),(86,109),(87,108),(88,107),(89,106),(90,105),(91,104),(92,103),(93,102),(94,101),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115)], [(1,33,103,53,149,68,98,123),(2,30,112,46,150,65,87,136),(3,27,101,59,151,62,96,129),(4,24,110,52,152,79,85,122),(5,21,119,45,153,76,94,135),(6,38,108,58,154,73,83,128),(7,35,117,51,155,70,92,121),(8,32,106,44,156,67,81,134),(9,29,115,57,157,64,90,127),(10,26,104,50,158,61,99,140),(11,23,113,43,159,78,88,133),(12,40,102,56,160,75,97,126),(13,37,111,49,141,72,86,139),(14,34,120,42,142,69,95,132),(15,31,109,55,143,66,84,125),(16,28,118,48,144,63,93,138),(17,25,107,41,145,80,82,131),(18,22,116,54,146,77,91,124),(19,39,105,47,147,74,100,137),(20,36,114,60,148,71,89,130)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 28 | 5 | 0 | 0 | 0 | 0 |
| 7 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 1 | 20 |
| 0 | 0 | 22 | 0 | 21 | 21 |
| 0 | 0 | 21 | 21 | 0 | 22 |
| 0 | 0 | 20 | 1 | 20 | 0 |
G:=sub<GL(6,GF(41))| [28,7,0,0,0,0,5,13,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1],[40,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0,0,1,0,40,0,0,0,1,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,22,21,20,0,0,20,0,21,1,0,0,1,21,0,20,0,0,20,21,22,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 | F5 | C2×F5 | D5⋊C8 | D4×F5 | Q8.F5 |
| kernel | D20⋊2C8 | C4×C5⋊C8 | C20⋊C8 | D10⋊C8 | D20⋊8C4 | C2×D5⋊C8 | D10⋊C4 | C5×C4⋊C4 | C2×D20 | D20 | C5⋊C8 | Dic5 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_2C_8 % in TeX
G:=Group("D20:2C8"); // GroupNames label
G:=SmallGroup(320,1040);
// by ID
G=gap.SmallGroup(320,1040);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations
×
⋊
ℤ
𝔽
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