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G = D4×C5⋊C8order 320 = 26·5

Direct product of D4 and C5⋊C8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×C5⋊C8, C53(C8×D4), C202(C2×C8), (C5×D4)⋊2C8, C2.5(D4×F5), C20⋊C85C2, (D4×C10).8C4, C10.26(C4×D4), (C2×D4).12F5, C4⋊Dic5.13C4, C2.5(D4.F5), C23.D5.5C4, C23.29(C2×F5), C10.14(C8○D4), C10.21(C22×C8), (D4×Dic5).17C2, Dic5.78(C2×D4), C23.2F58C2, Dic5.57(C4○D4), C22.51(C22×F5), (C4×Dic5).194C22, (C2×Dic5).351C23, (C22×Dic5).184C22, C41(C2×C5⋊C8), (C4×C5⋊C8)⋊5C2, C221(C2×C5⋊C8), (C2×C10)⋊2(C2×C8), (C22×C5⋊C8)⋊4C2, C2.6(C22×C5⋊C8), (C2×C4).80(C2×F5), (C2×C20).54(C2×C4), (C2×C5⋊C8).39C22, (C22×C10).23(C2×C4), (C2×C10).75(C22×C4), (C2×Dic5).70(C2×C4), SmallGroup(320,1110)

Series: Derived Chief Lower central Upper central

C1C10 — D4×C5⋊C8
C1C5C10Dic5C2×Dic5C2×C5⋊C8C22×C5⋊C8 — D4×C5⋊C8
C5C10 — D4×C5⋊C8
C1C22C2×D4

Generators and relations for D4×C5⋊C8
 G = < a,b,c,d | a4=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 394 in 134 conjugacy classes, 64 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C8×D4, C4×Dic5, C4⋊Dic5, C23.D5, C2×C5⋊C8, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5, D4×Dic5, C22×C5⋊C8, D4×C5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C5⋊C8, C2×F5, C8×D4, C2×C5⋊C8, C22×F5, D4.F5, D4×F5, C22×C5⋊C8, D4×C5⋊C8

Smallest permutation representation of D4×C5⋊C8
On 160 points
Generators in S160
(1 95 132 18)(2 96 133 19)(3 89 134 20)(4 90 135 21)(5 91 136 22)(6 92 129 23)(7 93 130 24)(8 94 131 17)(9 126 80 101)(10 127 73 102)(11 128 74 103)(12 121 75 104)(13 122 76 97)(14 123 77 98)(15 124 78 99)(16 125 79 100)(25 47 33 139)(26 48 34 140)(27 41 35 141)(28 42 36 142)(29 43 37 143)(30 44 38 144)(31 45 39 137)(32 46 40 138)(49 160 106 114)(50 153 107 115)(51 154 108 116)(52 155 109 117)(53 156 110 118)(54 157 111 119)(55 158 112 120)(56 159 105 113)(57 83 65 151)(58 84 66 152)(59 85 67 145)(60 86 68 146)(61 87 69 147)(62 88 70 148)(63 81 71 149)(64 82 72 150)
(17 94)(18 95)(19 96)(20 89)(21 90)(22 91)(23 92)(24 93)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(81 149)(82 150)(83 151)(84 152)(85 145)(86 146)(87 147)(88 148)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 121)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L 5 8A···8H8I···8T10A10B10C10D10E10F10G20A20B
order122222224444444···458···88···8101010101010102020
size1111222222555510···1045···510···10444888888

50 irreducible representations

dim1111111111222444488
type+++++++++-+-+
imageC1C2C2C2C2C2C4C4C4C8D4C4○D4C8○D4F5C2×F5C5⋊C8C2×F5D4.F5D4×F5
kernelD4×C5⋊C8C4×C5⋊C8C20⋊C8C23.2F5D4×Dic5C22×C5⋊C8C4⋊Dic5C23.D5D4×C10C5×D4C5⋊C8Dic5C10C2×D4C2×C4D4C23C2C2
# reps11121224216224114211

Matrix representation of D4×C5⋊C8 in GL8(𝔽41)

402000000
401000000
000400000
00100000
00001000
00000100
00000010
00000001
,
10000000
140000000
00100000
000400000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
270000000
027000000
00100000
00010000
00001511628
00002139282
0000213398
000013192630

G:=sub<GL(8,GF(41))| [40,40,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,21,2,13,0,0,0,0,11,39,13,19,0,0,0,0,6,28,39,26,0,0,0,0,28,2,8,30] >;

D4×C5⋊C8 in GAP, Magma, Sage, TeX

D_4\times C_5\rtimes C_8
% in TeX

G:=Group("D4xC5:C8");
// GroupNames label

G:=SmallGroup(320,1110);
// by ID

G=gap.SmallGroup(320,1110);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,6278,1595]);
// Polycyclic

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

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