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G = C202D8order 320 = 26·5

2nd semidirect product of C20 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202D8, D209D4, C42.73D10, C42(D4⋊D5), C41D41D5, C54(C4⋊D8), C4.53(D4×D5), (C4×D20)⋊22C2, C203C831C2, C20.30(C2×D4), C10.57(C2×D8), (C2×D4).55D10, (C2×C20).147D4, C20.76(C4○D4), D4⋊Dic522C2, C4.3(D42D5), C2.12(C202D4), C10.94(C8⋊C22), (C2×C20).390C23, (C4×C20).120C22, (D4×C10).71C22, C10.103(C4⋊D4), (C2×D20).254C22, C4⋊Dic5.344C22, C2.15(D4.D10), (C2×D4⋊D5)⋊13C2, (C5×C41D4)⋊1C2, C2.12(C2×D4⋊D5), (C2×C10).521(C2×D4), (C2×C4).185(C5⋊D4), (C2×C4).488(C22×D5), C22.194(C2×C5⋊D4), (C2×C52C8).130C22, SmallGroup(320,699)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C202D8
C1C5C10C20C2×C20C2×D20C4×D20 — C202D8
C5C10C2×C20 — C202D8
C1C22C42C41D4

Generators and relations for C202D8
 G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 582 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×10], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×11], C23 [×3], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×3], Dic5, C20 [×2], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8 [×2], C52C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C5×D4 [×8], C22×D5, C22×C10 [×2], C4⋊D8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C4×C20, C2×C4×D5, C2×D20, D4×C10 [×2], D4×C10 [×2], C203C8, D4⋊Dic5 [×2], C4×D20, C2×D4⋊D5 [×2], C5×C41D4, C202D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C4⋊D8, D4⋊D5 [×2], D4×D5, D42D5, C2×C5⋊D4, C2×D4⋊D5, D4.D10, C202D4, C202D8

Smallest permutation representation of C202D8
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 56 135 156 82 40 104 63)(2 55 136 155 83 39 105 62)(3 54 137 154 84 38 106 61)(4 53 138 153 85 37 107 80)(5 52 139 152 86 36 108 79)(6 51 140 151 87 35 109 78)(7 50 121 150 88 34 110 77)(8 49 122 149 89 33 111 76)(9 48 123 148 90 32 112 75)(10 47 124 147 91 31 113 74)(11 46 125 146 92 30 114 73)(12 45 126 145 93 29 115 72)(13 44 127 144 94 28 116 71)(14 43 128 143 95 27 117 70)(15 42 129 142 96 26 118 69)(16 41 130 141 97 25 119 68)(17 60 131 160 98 24 120 67)(18 59 132 159 99 23 101 66)(19 58 133 158 100 22 102 65)(20 57 134 157 81 21 103 64)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 145)(22 154)(23 143)(24 152)(25 141)(26 150)(27 159)(28 148)(29 157)(30 146)(31 155)(32 144)(33 153)(34 142)(35 151)(36 160)(37 149)(38 158)(39 147)(40 156)(41 68)(42 77)(43 66)(44 75)(45 64)(46 73)(47 62)(48 71)(49 80)(50 69)(51 78)(52 67)(53 76)(54 65)(55 74)(56 63)(57 72)(58 61)(59 70)(60 79)(81 93)(83 91)(84 100)(85 89)(86 98)(88 96)(90 94)(95 99)(101 128)(102 137)(103 126)(104 135)(105 124)(106 133)(107 122)(108 131)(109 140)(110 129)(111 138)(112 127)(113 136)(114 125)(115 134)(116 123)(117 132)(118 121)(119 130)(120 139)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G···10N20A···20L
order12222222444444455888810···1010···1020···20
size111188202022224202022202020202···28···84···4

47 irreducible representations

dim1111112222222244444
type+++++++++++++++-
imageC1C2C2C2C2C2D4D4D5D8C4○D4D10D10C5⋊D4C8⋊C22D4⋊D5D4×D5D42D5D4.D10
kernelC202D8C203C8D4⋊Dic5C4×D20C2×D4⋊D5C5×C41D4D20C2×C20C41D4C20C20C42C2×D4C2×C4C10C4C4C4C2
# reps1121212224224814224

Matrix representation of C202D8 in GL6(𝔽41)

100000
010000
001100
005600
00003833
0000323
,
17350000
700000
00353400
005600
0000215
00003520
,
1210000
0400000
006700
00363500
000010
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,0,38,32,0,0,0,0,33,3],[17,7,0,0,0,0,35,0,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,21,35,0,0,0,0,5,20],[1,0,0,0,0,0,21,40,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C202D8 in GAP, Magma, Sage, TeX

C_{20}\rtimes_2D_8
% in TeX

G:=Group("C20:2D8");
// GroupNames label

G:=SmallGroup(320,699);
// by ID

G=gap.SmallGroup(320,699);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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