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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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C41D4, C2×D8, C52C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊D8, C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C4×C20, C2×C4×D5, C2×D20, D4×C10, D4×C10, C203C8, D4⋊Dic5, C4×D20, C2×D4⋊D5, C5×C41D4, C202D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C4⋊D8, D4⋊D5, 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 64 44 139 32 120 95 157)(2 63 45 138 33 119 96 156)(3 62 46 137 34 118 97 155)(4 61 47 136 35 117 98 154)(5 80 48 135 36 116 99 153)(6 79 49 134 37 115 100 152)(7 78 50 133 38 114 81 151)(8 77 51 132 39 113 82 150)(9 76 52 131 40 112 83 149)(10 75 53 130 21 111 84 148)(11 74 54 129 22 110 85 147)(12 73 55 128 23 109 86 146)(13 72 56 127 24 108 87 145)(14 71 57 126 25 107 88 144)(15 70 58 125 26 106 89 143)(16 69 59 124 27 105 90 142)(17 68 60 123 28 104 91 141)(18 67 41 122 29 103 92 160)(19 66 42 121 30 102 93 159)(20 65 43 140 31 101 94 158)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)(41 88)(42 97)(43 86)(44 95)(45 84)(46 93)(47 82)(48 91)(49 100)(50 89)(51 98)(52 87)(53 96)(54 85)(55 94)(56 83)(57 92)(58 81)(59 90)(60 99)(61 150)(62 159)(63 148)(64 157)(65 146)(66 155)(67 144)(68 153)(69 142)(70 151)(71 160)(72 149)(73 158)(74 147)(75 156)(76 145)(77 154)(78 143)(79 152)(80 141)(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), (1,64,44,139,32,120,95,157)(2,63,45,138,33,119,96,156)(3,62,46,137,34,118,97,155)(4,61,47,136,35,117,98,154)(5,80,48,135,36,116,99,153)(6,79,49,134,37,115,100,152)(7,78,50,133,38,114,81,151)(8,77,51,132,39,113,82,150)(9,76,52,131,40,112,83,149)(10,75,53,130,21,111,84,148)(11,74,54,129,22,110,85,147)(12,73,55,128,23,109,86,146)(13,72,56,127,24,108,87,145)(14,71,57,126,25,107,88,144)(15,70,58,125,26,106,89,143)(16,69,59,124,27,105,90,142)(17,68,60,123,28,104,91,141)(18,67,41,122,29,103,92,160)(19,66,42,121,30,102,93,159)(20,65,43,140,31,101,94,158), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,88)(42,97)(43,86)(44,95)(45,84)(46,93)(47,82)(48,91)(49,100)(50,89)(51,98)(52,87)(53,96)(54,85)(55,94)(56,83)(57,92)(58,81)(59,90)(60,99)(61,150)(62,159)(63,148)(64,157)(65,146)(66,155)(67,144)(68,153)(69,142)(70,151)(71,160)(72,149)(73,158)(74,147)(75,156)(76,145)(77,154)(78,143)(79,152)(80,141)(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), (1,64,44,139,32,120,95,157)(2,63,45,138,33,119,96,156)(3,62,46,137,34,118,97,155)(4,61,47,136,35,117,98,154)(5,80,48,135,36,116,99,153)(6,79,49,134,37,115,100,152)(7,78,50,133,38,114,81,151)(8,77,51,132,39,113,82,150)(9,76,52,131,40,112,83,149)(10,75,53,130,21,111,84,148)(11,74,54,129,22,110,85,147)(12,73,55,128,23,109,86,146)(13,72,56,127,24,108,87,145)(14,71,57,126,25,107,88,144)(15,70,58,125,26,106,89,143)(16,69,59,124,27,105,90,142)(17,68,60,123,28,104,91,141)(18,67,41,122,29,103,92,160)(19,66,42,121,30,102,93,159)(20,65,43,140,31,101,94,158), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)(41,88)(42,97)(43,86)(44,95)(45,84)(46,93)(47,82)(48,91)(49,100)(50,89)(51,98)(52,87)(53,96)(54,85)(55,94)(56,83)(57,92)(58,81)(59,90)(60,99)(61,150)(62,159)(63,148)(64,157)(65,146)(66,155)(67,144)(68,153)(69,142)(70,151)(71,160)(72,149)(73,158)(74,147)(75,156)(76,145)(77,154)(78,143)(79,152)(80,141)(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)], [(1,64,44,139,32,120,95,157),(2,63,45,138,33,119,96,156),(3,62,46,137,34,118,97,155),(4,61,47,136,35,117,98,154),(5,80,48,135,36,116,99,153),(6,79,49,134,37,115,100,152),(7,78,50,133,38,114,81,151),(8,77,51,132,39,113,82,150),(9,76,52,131,40,112,83,149),(10,75,53,130,21,111,84,148),(11,74,54,129,22,110,85,147),(12,73,55,128,23,109,86,146),(13,72,56,127,24,108,87,145),(14,71,57,126,25,107,88,144),(15,70,58,125,26,106,89,143),(16,69,59,124,27,105,90,142),(17,68,60,123,28,104,91,141),(18,67,41,122,29,103,92,160),(19,66,42,121,30,102,93,159),(20,65,43,140,31,101,94,158)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39),(41,88),(42,97),(43,86),(44,95),(45,84),(46,93),(47,82),(48,91),(49,100),(50,89),(51,98),(52,87),(53,96),(54,85),(55,94),(56,83),(57,92),(58,81),(59,90),(60,99),(61,150),(62,159),(63,148),(64,157),(65,146),(66,155),(67,144),(68,153),(69,142),(70,151),(71,160),(72,149),(73,158),(74,147),(75,156),(76,145),(77,154),(78,143),(79,152),(80,141),(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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