Copied to
clipboard

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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊D4, C8⋊D5, C2×C52C8, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D4.D5, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C22×D5, C8⋊D4, C4○D20, D4×D5, D10⋊D4, D8⋊D5, SD16⋊D5, C52C8⋊D4

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

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

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

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

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

׿
×
𝔽