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G = D202C8order 320 = 26·5

2nd semidirect product of D20 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D202C8, C5⋊C89D4, C52(C8×D4), C201(C2×C8), C2.3(D4×F5), C41(D5⋊C8), D102(C2×C8), C20⋊C83C2, C10.8(C4×D4), C4⋊C4.12F5, (C2×D20).11C4, C10.6(C22×C8), D10⋊C813C2, C2.2(Q8.F5), D10⋊C4.8C4, C10.19(C8○D4), Dic5.69(C2×D4), D208C4.16C2, Dic5.54(C4○D4), C22.37(C22×F5), (C2×Dic5).329C23, (C4×Dic5).190C22, (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

C1C10 — D202C8
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×D5⋊C8 — D202C8
C5C10 — D202C8
C1C22C4⋊C4

Generators and relations for D202C8
 G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a12b >

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], D208C4, C2×D5⋊C8 [×2], D202C8
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, D202C8

Smallest permutation representation of D202C8
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 110)(38 109)(39 108)(40 107)(61 143)(62 142)(63 141)(64 160)(65 159)(66 158)(67 157)(68 156)(69 155)(70 154)(71 153)(72 152)(73 151)(74 150)(75 149)(76 148)(77 147)(78 146)(79 145)(80 144)(81 122)(82 121)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)
(1 127 22 158 46 82 111 72)(2 124 31 151 47 99 120 65)(3 121 40 144 48 96 109 78)(4 138 29 157 49 93 118 71)(5 135 38 150 50 90 107 64)(6 132 27 143 51 87 116 77)(7 129 36 156 52 84 105 70)(8 126 25 149 53 81 114 63)(9 123 34 142 54 98 103 76)(10 140 23 155 55 95 112 69)(11 137 32 148 56 92 101 62)(12 134 21 141 57 89 110 75)(13 131 30 154 58 86 119 68)(14 128 39 147 59 83 108 61)(15 125 28 160 60 100 117 74)(16 122 37 153 41 97 106 67)(17 139 26 146 42 94 115 80)(18 136 35 159 43 91 104 73)(19 133 24 152 44 88 113 66)(20 130 33 145 45 85 102 79)

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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(61,143)(62,142)(63,141)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,154)(71,153)(72,152)(73,151)(74,150)(75,149)(76,148)(77,147)(78,146)(79,145)(80,144)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123), (1,127,22,158,46,82,111,72)(2,124,31,151,47,99,120,65)(3,121,40,144,48,96,109,78)(4,138,29,157,49,93,118,71)(5,135,38,150,50,90,107,64)(6,132,27,143,51,87,116,77)(7,129,36,156,52,84,105,70)(8,126,25,149,53,81,114,63)(9,123,34,142,54,98,103,76)(10,140,23,155,55,95,112,69)(11,137,32,148,56,92,101,62)(12,134,21,141,57,89,110,75)(13,131,30,154,58,86,119,68)(14,128,39,147,59,83,108,61)(15,125,28,160,60,100,117,74)(16,122,37,153,41,97,106,67)(17,139,26,146,42,94,115,80)(18,136,35,159,43,91,104,73)(19,133,24,152,44,88,113,66)(20,130,33,145,45,85,102,79)>;

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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(61,143)(62,142)(63,141)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,154)(71,153)(72,152)(73,151)(74,150)(75,149)(76,148)(77,147)(78,146)(79,145)(80,144)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123), (1,127,22,158,46,82,111,72)(2,124,31,151,47,99,120,65)(3,121,40,144,48,96,109,78)(4,138,29,157,49,93,118,71)(5,135,38,150,50,90,107,64)(6,132,27,143,51,87,116,77)(7,129,36,156,52,84,105,70)(8,126,25,149,53,81,114,63)(9,123,34,142,54,98,103,76)(10,140,23,155,55,95,112,69)(11,137,32,148,56,92,101,62)(12,134,21,141,57,89,110,75)(13,131,30,154,58,86,119,68)(14,128,39,147,59,83,108,61)(15,125,28,160,60,100,117,74)(16,122,37,153,41,97,106,67)(17,139,26,146,42,94,115,80)(18,136,35,159,43,91,104,73)(19,133,24,152,44,88,113,66)(20,130,33,145,45,85,102,79) );

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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,106),(22,105),(23,104),(24,103),(25,102),(26,101),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,110),(38,109),(39,108),(40,107),(61,143),(62,142),(63,141),(64,160),(65,159),(66,158),(67,157),(68,156),(69,155),(70,154),(71,153),(72,152),(73,151),(74,150),(75,149),(76,148),(77,147),(78,146),(79,145),(80,144),(81,122),(82,121),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123)], [(1,127,22,158,46,82,111,72),(2,124,31,151,47,99,120,65),(3,121,40,144,48,96,109,78),(4,138,29,157,49,93,118,71),(5,135,38,150,50,90,107,64),(6,132,27,143,51,87,116,77),(7,129,36,156,52,84,105,70),(8,126,25,149,53,81,114,63),(9,123,34,142,54,98,103,76),(10,140,23,155,55,95,112,69),(11,137,32,148,56,92,101,62),(12,134,21,141,57,89,110,75),(13,131,30,154,58,86,119,68),(14,128,39,147,59,83,108,61),(15,125,28,160,60,100,117,74),(16,122,37,153,41,97,106,67),(17,139,26,146,42,94,115,80),(18,136,35,159,43,91,104,73),(19,133,24,152,44,88,113,66),(20,130,33,145,45,85,102,79)])

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J4K4L 5 8A···8H8I···8T10A10B10C20A···20F
order122222224···444444458···88···810101020···20
size1111101010102···25555101045···510···104448···8

50 irreducible representations

dim111111111122244488
type+++++++++++
imageC1C2C2C2C2C2C4C4C4C8D4C4○D4C8○D4F5C2×F5D5⋊C8D4×F5Q8.F5
kernelD202C8C4×C5⋊C8C20⋊C8D10⋊C8D208C4C2×D5⋊C8D10⋊C4C5×C4⋊C4C2×D20D20C5⋊C8Dic5C10C4⋊C4C2×C4C4C2C2
# reps1112124221622413411

Matrix representation of D202C8 in GL6(𝔽41)

2850000
7130000
0004000
0000400
0000040
001111
,
4000000
310000
001111
0000040
0000400
0004000
,
4000000
0400000
00020120
002202121
002121022
00201200

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] >;

D202C8 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

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