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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 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C8 [×5], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C8×D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×4], C22×Dic5 [×2], D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5 [×2], D4×Dic5, C22×C5⋊C8 [×2], D4×C5⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C5⋊C8 [×4], C2×F5 [×3], C8×D4, C2×C5⋊C8 [×6], 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 71 57 83)(2 72 58 84)(3 65 59 85)(4 66 60 86)(5 67 61 87)(6 68 62 88)(7 69 63 81)(8 70 64 82)(9 130 24 89)(10 131 17 90)(11 132 18 91)(12 133 19 92)(13 134 20 93)(14 135 21 94)(15 136 22 95)(16 129 23 96)(25 151 125 79)(26 152 126 80)(27 145 127 73)(28 146 128 74)(29 147 121 75)(30 148 122 76)(31 149 123 77)(32 150 124 78)(33 143 104 43)(34 144 97 44)(35 137 98 45)(36 138 99 46)(37 139 100 47)(38 140 101 48)(39 141 102 41)(40 142 103 42)(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)
(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 125)(26 126)(27 127)(28 128)(29 121)(30 122)(31 123)(32 124)(33 104)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)
(1 147 132 143 113)(2 144 148 114 133)(3 115 137 134 149)(4 135 116 150 138)(5 151 136 139 117)(6 140 152 118 129)(7 119 141 130 145)(8 131 120 146 142)(9 27 81 111 39)(10 112 28 40 82)(11 33 105 83 29)(12 84 34 30 106)(13 31 85 107 35)(14 108 32 36 86)(15 37 109 87 25)(16 88 38 26 110)(17 55 128 103 70)(18 104 56 71 121)(19 72 97 122 49)(20 123 65 50 98)(21 51 124 99 66)(22 100 52 67 125)(23 68 101 126 53)(24 127 69 54 102)(41 89 73 63 157)(42 64 90 158 74)(43 159 57 75 91)(44 76 160 92 58)(45 93 77 59 153)(46 60 94 154 78)(47 155 61 79 95)(48 80 156 96 62)
(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)| (1,71,57,83)(2,72,58,84)(3,65,59,85)(4,66,60,86)(5,67,61,87)(6,68,62,88)(7,69,63,81)(8,70,64,82)(9,130,24,89)(10,131,17,90)(11,132,18,91)(12,133,19,92)(13,134,20,93)(14,135,21,94)(15,136,22,95)(16,129,23,96)(25,151,125,79)(26,152,126,80)(27,145,127,73)(28,146,128,74)(29,147,121,75)(30,148,122,76)(31,149,123,77)(32,150,124,78)(33,143,104,43)(34,144,97,44)(35,137,98,45)(36,138,99,46)(37,139,100,47)(38,140,101,48)(39,141,102,41)(40,142,103,42)(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), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,104)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84), (1,147,132,143,113)(2,144,148,114,133)(3,115,137,134,149)(4,135,116,150,138)(5,151,136,139,117)(6,140,152,118,129)(7,119,141,130,145)(8,131,120,146,142)(9,27,81,111,39)(10,112,28,40,82)(11,33,105,83,29)(12,84,34,30,106)(13,31,85,107,35)(14,108,32,36,86)(15,37,109,87,25)(16,88,38,26,110)(17,55,128,103,70)(18,104,56,71,121)(19,72,97,122,49)(20,123,65,50,98)(21,51,124,99,66)(22,100,52,67,125)(23,68,101,126,53)(24,127,69,54,102)(41,89,73,63,157)(42,64,90,158,74)(43,159,57,75,91)(44,76,160,92,58)(45,93,77,59,153)(46,60,94,154,78)(47,155,61,79,95)(48,80,156,96,62), (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( (1,71,57,83)(2,72,58,84)(3,65,59,85)(4,66,60,86)(5,67,61,87)(6,68,62,88)(7,69,63,81)(8,70,64,82)(9,130,24,89)(10,131,17,90)(11,132,18,91)(12,133,19,92)(13,134,20,93)(14,135,21,94)(15,136,22,95)(16,129,23,96)(25,151,125,79)(26,152,126,80)(27,145,127,73)(28,146,128,74)(29,147,121,75)(30,148,122,76)(31,149,123,77)(32,150,124,78)(33,143,104,43)(34,144,97,44)(35,137,98,45)(36,138,99,46)(37,139,100,47)(38,140,101,48)(39,141,102,41)(40,142,103,42)(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), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,104)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,105)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84), (1,147,132,143,113)(2,144,148,114,133)(3,115,137,134,149)(4,135,116,150,138)(5,151,136,139,117)(6,140,152,118,129)(7,119,141,130,145)(8,131,120,146,142)(9,27,81,111,39)(10,112,28,40,82)(11,33,105,83,29)(12,84,34,30,106)(13,31,85,107,35)(14,108,32,36,86)(15,37,109,87,25)(16,88,38,26,110)(17,55,128,103,70)(18,104,56,71,121)(19,72,97,122,49)(20,123,65,50,98)(21,51,124,99,66)(22,100,52,67,125)(23,68,101,126,53)(24,127,69,54,102)(41,89,73,63,157)(42,64,90,158,74)(43,159,57,75,91)(44,76,160,92,58)(45,93,77,59,153)(46,60,94,154,78)(47,155,61,79,95)(48,80,156,96,62), (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([(1,71,57,83),(2,72,58,84),(3,65,59,85),(4,66,60,86),(5,67,61,87),(6,68,62,88),(7,69,63,81),(8,70,64,82),(9,130,24,89),(10,131,17,90),(11,132,18,91),(12,133,19,92),(13,134,20,93),(14,135,21,94),(15,136,22,95),(16,129,23,96),(25,151,125,79),(26,152,126,80),(27,145,127,73),(28,146,128,74),(29,147,121,75),(30,148,122,76),(31,149,123,77),(32,150,124,78),(33,143,104,43),(34,144,97,44),(35,137,98,45),(36,138,99,46),(37,139,100,47),(38,140,101,48),(39,141,102,41),(40,142,103,42),(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)], [(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,125),(26,126),(27,127),(28,128),(29,121),(30,122),(31,123),(32,124),(33,104),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,105),(65,85),(66,86),(67,87),(68,88),(69,81),(70,82),(71,83),(72,84)], [(1,147,132,143,113),(2,144,148,114,133),(3,115,137,134,149),(4,135,116,150,138),(5,151,136,139,117),(6,140,152,118,129),(7,119,141,130,145),(8,131,120,146,142),(9,27,81,111,39),(10,112,28,40,82),(11,33,105,83,29),(12,84,34,30,106),(13,31,85,107,35),(14,108,32,36,86),(15,37,109,87,25),(16,88,38,26,110),(17,55,128,103,70),(18,104,56,71,121),(19,72,97,122,49),(20,123,65,50,98),(21,51,124,99,66),(22,100,52,67,125),(23,68,101,126,53),(24,127,69,54,102),(41,89,73,63,157),(42,64,90,158,74),(43,159,57,75,91),(44,76,160,92,58),(45,93,77,59,153),(46,60,94,154,78),(47,155,61,79,95),(48,80,156,96,62)], [(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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