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G = C52C8⋊D4order 320 = 26·5

1st semidirect product of C52C8 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C81D4, C51(C8⋊D4), C4⋊C4.11D10, C4.160(D4×D5), D4⋊C418D5, D102Q83C2, (C2×D4).28D10, C20.9(C4○D4), C10.D88C2, C202D4.6C2, C20.110(C2×D4), (C2×C8).169D10, C4.26(C4○D20), (C2×Dic5).30D4, (C22×D5).20D4, C22.178(D4×D5), C20.44D421C2, C2.16(D8⋊D5), C10.17(C4⋊D4), C10.34(C8⋊C22), (C2×C20).220C23, (C2×C40).186C22, (D4×C10).41C22, C4⋊Dic5.74C22, C2.20(D10⋊D4), C2.12(SD16⋊D5), C10.30(C8.C22), (C2×Dic10).62C22, (C2×D4.D5)⋊5C2, (C2×C8⋊D5)⋊17C2, (C5×D4⋊C4)⋊24C2, (C2×C4×D5).16C22, (C2×C10).233(C2×D4), (C5×C4⋊C4).21C22, (C2×C52C8).18C22, (C2×C4).327(C22×D5), SmallGroup(320,407)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C52C8⋊D4
C1C5C10C20C2×C20C2×C4×D5C202D4 — C52C8⋊D4
C5C10C2×C20 — C52C8⋊D4
C1C22C2×C4D4⋊C4

Generators and relations for C52C8⋊D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=dad=a-1, cbc-1=b-1, dbd=b5, dcd=c-1 >

Subgroups: 518 in 120 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×3], C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5, C10 [×3], C10, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8 [×2], C40, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C8⋊D4, C8⋊D5 [×2], C2×C52C8, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D4.D5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×C5⋊D4, D4×C10, C10.D8, C20.44D4, C5×D4⋊C4, D102Q8, C2×C8⋊D5, C2×D4.D5, C202D4, C52C8⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C22×D5, C8⋊D4, C4○D20, D4×D5 [×2], D10⋊D4, D8⋊D5, SD16⋊D5, C52C8⋊D4

Smallest permutation representation of C52C8⋊D4
On 160 points
Generators in S160
(1 105 35 97 149)(2 150 98 36 106)(3 107 37 99 151)(4 152 100 38 108)(5 109 39 101 145)(6 146 102 40 110)(7 111 33 103 147)(8 148 104 34 112)(9 58 67 128 56)(10 49 121 68 59)(11 60 69 122 50)(12 51 123 70 61)(13 62 71 124 52)(14 53 125 72 63)(15 64 65 126 54)(16 55 127 66 57)(17 90 159 85 113)(18 114 86 160 91)(19 92 153 87 115)(20 116 88 154 93)(21 94 155 81 117)(22 118 82 156 95)(23 96 157 83 119)(24 120 84 158 89)(25 79 139 47 133)(26 134 48 140 80)(27 73 141 41 135)(28 136 42 142 74)(29 75 143 43 129)(30 130 44 144 76)(31 77 137 45 131)(32 132 46 138 78)
(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 93 44 121)(2 92 45 128)(3 91 46 127)(4 90 47 126)(5 89 48 125)(6 96 41 124)(7 95 42 123)(8 94 43 122)(9 36 87 77)(10 35 88 76)(11 34 81 75)(12 33 82 74)(13 40 83 73)(14 39 84 80)(15 38 85 79)(16 37 86 78)(17 133 65 152)(18 132 66 151)(19 131 67 150)(20 130 68 149)(21 129 69 148)(22 136 70 147)(23 135 71 146)(24 134 72 145)(25 64 100 113)(26 63 101 120)(27 62 102 119)(28 61 103 118)(29 60 104 117)(30 59 97 116)(31 58 98 115)(32 57 99 114)(49 105 154 144)(50 112 155 143)(51 111 156 142)(52 110 157 141)(53 109 158 140)(54 108 159 139)(55 107 160 138)(56 106 153 137)
(2 6)(4 8)(9 119)(10 116)(11 113)(12 118)(13 115)(14 120)(15 117)(16 114)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(25 75)(26 80)(27 77)(28 74)(29 79)(30 76)(31 73)(32 78)(33 103)(34 100)(35 97)(36 102)(37 99)(38 104)(39 101)(40 98)(41 45)(43 47)(57 86)(58 83)(59 88)(60 85)(61 82)(62 87)(63 84)(64 81)(65 155)(66 160)(67 157)(68 154)(69 159)(70 156)(71 153)(72 158)(89 125)(90 122)(91 127)(92 124)(93 121)(94 126)(95 123)(96 128)(105 149)(106 146)(107 151)(108 148)(109 145)(110 150)(111 147)(112 152)(129 139)(130 144)(131 141)(132 138)(133 143)(134 140)(135 137)(136 142)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444455888810···1010101010202020202020202040···40
size1111820228204040224420202···28888444488884···4

44 irreducible representations

dim11111111222222222444444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C4○D20C8⋊C22C8.C22D4×D5D4×D5D8⋊D5SD16⋊D5
kernelC52C8⋊D4C10.D8C20.44D4C5×D4⋊C4D102Q8C2×C8⋊D5C2×D4.D5C202D4C52C8C2×Dic5C22×D5D4⋊C4C20C4⋊C4C2×C8C2×D4C4C10C10C4C22C2C2
# reps11111111211222228112244

Matrix representation of C52C8⋊D4 in GL10(𝔽41)

1000000000
0100000000
003440000000
0010000000
000034400000
0000100000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
00320000000
00229000000
0000900000
000019320000
0000001721122
0000002325039
00000030101723
00000012213223
,
04000000000
1000000000
0017401800000
0032438230000
0032017400000
002293240000
00000024900
00000091700
0000001016327
000000404129
,
40000000000
0100000000
0010000000
003440000000
00119100000
00143034400000
0000001000
0000000100
0000003423400
0000002325040

G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,32,22,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,19,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,17,23,30,12,0,0,0,0,0,0,21,25,10,21,0,0,0,0,0,0,1,0,17,32,0,0,0,0,0,0,22,39,23,23],[0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,17,3,32,22,0,0,0,0,0,0,40,24,0,9,0,0,0,0,0,0,18,38,17,3,0,0,0,0,0,0,0,23,40,24,0,0,0,0,0,0,0,0,0,0,24,9,10,40,0,0,0,0,0,0,9,17,16,4,0,0,0,0,0,0,0,0,32,12,0,0,0,0,0,0,0,0,7,9],[40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,34,11,14,0,0,0,0,0,0,0,40,9,30,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,34,23,0,0,0,0,0,0,0,1,23,25,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40] >;

C52C8⋊D4 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8\rtimes D_4
% in TeX

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

G:=SmallGroup(320,407);
// by ID

G=gap.SmallGroup(320,407);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,1094,135,100,570,297,136,12550]);
// Polycyclic

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

׿
×
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