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## G = Dic5⋊D8order 320 = 26·5

### 2nd semidirect product of Dic5 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5⋊D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — D4×Dic5 — Dic5⋊D8
 Lower central C5 — C10 — C2×C20 — Dic5⋊D8
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for Dic5⋊D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >

Subgroups: 662 in 140 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×9], C23 [×3], D5, C10 [×3], C10 [×3], C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×7], D4⋊C4 [×2], C4⋊C8, C4×D4, C41D4, C2×D8, C2×D8, C52C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C5×D4 [×3], C22×D5, C22×C10 [×2], C4⋊D8, C2×C52C8, C4×Dic5, C4⋊Dic5, D4⋊D5 [×2], C23.D5, C2×C40, C5×D8 [×2], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], D4×C10 [×2], C20.8Q8, D205C4, D4⋊Dic5, C2×D4⋊D5, D4×Dic5, C20⋊D4, C10×D8, Dic5⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C4⋊D8, D4×D5, D42D5, C2×C5⋊D4, D5×D8, D8⋊D5, Dic5⋊D4, Dic5⋊D8

Smallest permutation representation of Dic5⋊D8
On 160 points
Generators in S160
(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 54 6 59)(2 53 7 58)(3 52 8 57)(4 51 9 56)(5 60 10 55)(11 112 16 117)(12 111 17 116)(13 120 18 115)(14 119 19 114)(15 118 20 113)(21 65 26 70)(22 64 27 69)(23 63 28 68)(24 62 29 67)(25 61 30 66)(31 81 36 86)(32 90 37 85)(33 89 38 84)(34 88 39 83)(35 87 40 82)(41 71 46 76)(42 80 47 75)(43 79 48 74)(44 78 49 73)(45 77 50 72)(91 137 96 132)(92 136 97 131)(93 135 98 140)(94 134 99 139)(95 133 100 138)(101 145 106 150)(102 144 107 149)(103 143 108 148)(104 142 109 147)(105 141 110 146)(121 151 126 156)(122 160 127 155)(123 159 128 154)(124 158 129 153)(125 157 130 152)
(1 154 34 148 28 14 48 134)(2 155 35 149 29 15 49 135)(3 156 36 150 30 16 50 136)(4 157 37 141 21 17 41 137)(5 158 38 142 22 18 42 138)(6 159 39 143 23 19 43 139)(7 160 40 144 24 20 44 140)(8 151 31 145 25 11 45 131)(9 152 32 146 26 12 46 132)(10 153 33 147 27 13 47 133)(51 125 85 105 65 111 71 91)(52 126 86 106 66 112 72 92)(53 127 87 107 67 113 73 93)(54 128 88 108 68 114 74 94)(55 129 89 109 69 115 75 95)(56 130 90 110 70 116 76 96)(57 121 81 101 61 117 77 97)(58 122 82 102 62 118 78 98)(59 123 83 103 63 119 79 99)(60 124 84 104 64 120 80 100)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 8 40 2 2 10 10 20 20 20 2 2 4 4 20 20 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D8 C4○D4 D10 D10 C5⋊D4 C8⋊C22 D4⋊2D5 D4×D5 D5×D8 D8⋊D5 kernel Dic5⋊D8 C20.8Q8 D20⋊5C4 D4⋊Dic5 C2×D4⋊D5 D4×Dic5 C20⋊D4 C10×D8 C2×Dic5 C5×D4 C2×D8 Dic5 C20 C2×C8 C2×D4 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 4 8 1 2 2 4 4

Matrix representation of Dic5⋊D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 15 4 0 0 0 0 5 26 0 0 0 0 0 0 34 35 0 0 0 0 8 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 37 0 0 0 0 15 15 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 22 0 0 0 0 3 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 22 0 0 0 0 4 35

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,3,0,0,0,0,22,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,4,0,0,0,0,22,35] >;

Dic5⋊D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes D_8
% in TeX

G:=Group("Dic5:D8");
// GroupNames label

G:=SmallGroup(320,777);
// by ID

G=gap.SmallGroup(320,777);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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