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G = C5⋊(C82D4)  order 320 = 26·5

The semidirect product of C5 and C82D4 acting via C82D4/D4⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C82D4, C51(C82D4), C202D42C2, C4⋊D204C2, C4⋊C4.13D10, C4.161(D4×D5), D4⋊C419D5, (C2×D4).29D10, (C2×C8).170D10, C20.111(C2×D4), C20.Q86C2, D205C420C2, C4.27(C4○D20), C20.10(C4○D4), (C2×Dic5).31D4, (C22×D5).22D4, C22.180(D4×D5), C2.18(D8⋊D5), C10.18(C4⋊D4), C2.11(D40⋊C2), C10.36(C8⋊C22), (C2×C20).222C23, (C2×C40).187C22, (C2×D20).56C22, (D4×C10).43C22, C4⋊Dic5.75C22, C2.21(D10⋊D4), (C2×D4⋊D5)⋊5C2, (C2×C8⋊D5)⋊18C2, (C5×D4⋊C4)⋊25C2, (C2×C4×D5).18C22, (C2×C10).235(C2×D4), (C5×C4⋊C4).23C22, (C2×C52C8).20C22, (C2×C4).329(C22×D5), SmallGroup(320,409)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C5⋊(C82D4)
C1C5C10C20C2×C20C2×C4×D5C4⋊D20 — C5⋊(C82D4)
C5C10C2×C20 — C5⋊(C82D4)
C1C22C2×C4D4⋊C4

Generators and relations for C5⋊(C82D4)
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 662 in 130 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×3], C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5 [×2], C10 [×3], C10, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2) [×2], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×3], D4⋊C4, D4⋊C4, C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C52C8 [×2], C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×D5, C22×C10, C82D4, C8⋊D5 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, D4×C10, C20.Q8, D205C4, C5×D4⋊C4, C4⋊D20, C2×C8⋊D5, C2×D4⋊D5, C202D4, C5⋊(C82D4)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], C22×D5, C82D4, C4○D20, D4×D5 [×2], D10⋊D4, D8⋊D5, D40⋊C2, C5⋊(C82D4)

Smallest permutation representation of C5⋊(C82D4)
On 160 points
Generators in S160
(1 74 83 143 10)(2 11 144 84 75)(3 76 85 137 12)(4 13 138 86 77)(5 78 87 139 14)(6 15 140 88 79)(7 80 81 141 16)(8 9 142 82 73)(17 129 45 147 118)(18 119 148 46 130)(19 131 47 149 120)(20 113 150 48 132)(21 133 41 151 114)(22 115 152 42 134)(23 135 43 145 116)(24 117 146 44 136)(25 108 59 101 36)(26 37 102 60 109)(27 110 61 103 38)(28 39 104 62 111)(29 112 63 97 40)(30 33 98 64 105)(31 106 57 99 34)(32 35 100 58 107)(49 124 159 70 95)(50 96 71 160 125)(51 126 153 72 89)(52 90 65 154 127)(53 128 155 66 91)(54 92 67 156 121)(55 122 157 68 93)(56 94 69 158 123)
(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)
(1 107 41 65)(2 110 42 68)(3 105 43 71)(4 108 44 66)(5 111 45 69)(6 106 46 72)(7 109 47 67)(8 112 48 70)(9 63 132 95)(10 58 133 90)(11 61 134 93)(12 64 135 96)(13 59 136 91)(14 62 129 94)(15 57 130 89)(16 60 131 92)(17 56 139 104)(18 51 140 99)(19 54 141 102)(20 49 142 97)(21 52 143 100)(22 55 144 103)(23 50 137 98)(24 53 138 101)(25 146 155 77)(26 149 156 80)(27 152 157 75)(28 147 158 78)(29 150 159 73)(30 145 160 76)(31 148 153 79)(32 151 154 74)(33 116 125 85)(34 119 126 88)(35 114 127 83)(36 117 128 86)(37 120 121 81)(38 115 122 84)(39 118 123 87)(40 113 124 82)
(1 45)(2 44)(3 43)(4 42)(5 41)(6 48)(7 47)(8 46)(9 148)(10 147)(11 146)(12 145)(13 152)(14 151)(15 150)(16 149)(17 83)(18 82)(19 81)(20 88)(21 87)(22 86)(23 85)(24 84)(25 61)(26 60)(27 59)(28 58)(29 57)(30 64)(31 63)(32 62)(33 98)(34 97)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 128)(56 127)(65 69)(66 68)(70 72)(73 130)(74 129)(75 136)(76 135)(77 134)(78 133)(79 132)(80 131)(89 159)(90 158)(91 157)(92 156)(93 155)(94 154)(95 153)(96 160)(106 112)(107 111)(108 110)(113 140)(114 139)(115 138)(116 137)(117 144)(118 143)(119 142)(120 141)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444455888810···1010101010202020202020202040···40
size1111820402282040224420202···28888444488884···4

44 irreducible representations

dim1111111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C4○D20C8⋊C22D4×D5D4×D5D8⋊D5D40⋊C2
kernelC5⋊(C82D4)C20.Q8D205C4C5×D4⋊C4C4⋊D20C2×C8⋊D5C2×D4⋊D5C202D4C52C8C2×Dic5C22×D5D4⋊C4C20C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps1111111121122222822244

Matrix representation of C5⋊(C82D4) in GL6(𝔽41)

100000
010000
000100
0040600
000001
0000406
,
3750000
1340000
00300629
0016112435
003863629
00293405
,
540000
14360000
00735220
00612022
001835346
006233529
,
4000000
2310000
001000
0064000
0010400
00640351

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[37,13,0,0,0,0,5,4,0,0,0,0,0,0,30,16,38,29,0,0,0,11,6,3,0,0,6,24,36,40,0,0,29,35,29,5],[5,14,0,0,0,0,4,36,0,0,0,0,0,0,7,6,18,6,0,0,35,12,35,23,0,0,22,0,34,35,0,0,0,22,6,29],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,1,6,1,6,0,0,0,40,0,40,0,0,0,0,40,35,0,0,0,0,0,1] >;

C5⋊(C82D4) in GAP, Magma, Sage, TeX

C_5\rtimes (C_8\rtimes_2D_4)
% in TeX

G:=Group("C5:(C8:2D4)");
// GroupNames label

G:=SmallGroup(320,409);
// by ID

G=gap.SmallGroup(320,409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽