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## G = D4×C5⋊C8order 320 = 26·5

### Direct product of D4 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — D4×C5⋊C8
 Lower central C5 — C10 — D4×C5⋊C8
 Upper central C1 — C22 — C2×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 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4L 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 8 ··· 8 8 ··· 8 10 10 10 10 10 10 10 20 20 size 1 1 1 1 2 2 2 2 2 2 5 5 5 5 10 ··· 10 4 5 ··· 5 10 ··· 10 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 8 type + + + + + + + + + - + - + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 D4 C4○D4 C8○D4 F5 C2×F5 C5⋊C8 C2×F5 D4.F5 D4×F5 kernel D4×C5⋊C8 C4×C5⋊C8 C20⋊C8 C23.2F5 D4×Dic5 C22×C5⋊C8 C4⋊Dic5 C23.D5 D4×C10 C5×D4 C5⋊C8 Dic5 C10 C2×D4 C2×C4 D4 C23 C2 C2 # reps 1 1 1 2 1 2 2 4 2 16 2 2 4 1 1 4 2 1 1

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

 40 2 0 0 0 0 0 0 40 1 0 0 0 0 0 0 0 0 0 40 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 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 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 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 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 11 6 28 0 0 0 0 21 39 28 2 0 0 0 0 2 13 39 8 0 0 0 0 13 19 26 30

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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