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## G = (C8×Dic5)⋊C2order 320 = 26·5

### 20th semidirect product of C8×Dic5 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C8×Dic5)⋊C2
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C20.17D4 — (C8×Dic5)⋊C2
 Lower central C5 — C10 — C2×C20 — (C8×Dic5)⋊C2
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for (C8×Dic5)⋊C2
G = < a,b,c,d | a8=b10=d2=1, c2=b5, ab=ba, ac=ca, dad=a-1b5, cbc-1=b-1, bd=db, dcd=a4b5c >

Subgroups: 350 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×2], C23, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4 [×2], C4.4D4, C42.C2, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×C10, C42.78C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, D4×C10, C10.Q16, C8×Dic5, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, C20.17D4, (C8×Dic5)⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C4○D8 [×2], C22×D5, C42.78C22, C4○D20, D4×D5, D42D5, Dic5.5D4, D83D5, SD163D5, (C8×Dic5)⋊C2

Smallest permutation representation of (C8×Dic5)⋊C2
On 160 points
Generators in S160
(1 60 35 85 27 67 42 75)(2 51 36 86 28 68 43 76)(3 52 37 87 29 69 44 77)(4 53 38 88 30 70 45 78)(5 54 39 89 21 61 46 79)(6 55 40 90 22 62 47 80)(7 56 31 81 23 63 48 71)(8 57 32 82 24 64 49 72)(9 58 33 83 25 65 50 73)(10 59 34 84 26 66 41 74)(11 103 143 128 156 91 131 116)(12 104 144 129 157 92 132 117)(13 105 145 130 158 93 133 118)(14 106 146 121 159 94 134 119)(15 107 147 122 160 95 135 120)(16 108 148 123 151 96 136 111)(17 109 149 124 152 97 137 112)(18 110 150 125 153 98 138 113)(19 101 141 126 154 99 139 114)(20 102 142 127 155 100 140 115)
(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 91 6 96)(2 100 7 95)(3 99 8 94)(4 98 9 93)(5 97 10 92)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 120 36 115)(32 119 37 114)(33 118 38 113)(34 117 39 112)(35 116 40 111)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 140 56 135)(52 139 57 134)(53 138 58 133)(54 137 59 132)(55 136 60 131)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 160 76 155)(72 159 77 154)(73 158 78 153)(74 157 79 152)(75 156 80 151)
(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 D4⋊2D5 D4×D5 D8⋊3D5 SD16⋊3D5 kernel (C8×Dic5)⋊C2 C10.Q16 C8×Dic5 C20.44D4 D4⋊Dic5 C5×D4⋊C4 C4.Dic10 C20.17D4 C2×Dic5 D4⋊C4 C20 C4⋊C4 C2×C8 C2×D4 C10 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 4 2 2 2 8 8 2 2 4 4

Matrix representation of (C8×Dic5)⋊C2 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 30 30 0 0 0 0 26 0
,
 40 40 0 0 0 0 8 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 38 33 0 0 0 0 1 3 0 0 0 0 0 0 7 18 0 0 0 0 20 34 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 22 40 0 0 0 0 0 0 1 0 0 0 0 0 40 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[40,8,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[38,1,0,0,0,0,33,3,0,0,0,0,0,0,7,20,0,0,0,0,18,34,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,22,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;

(C8×Dic5)⋊C2 in GAP, Magma, Sage, TeX

(C_8\times {\rm Dic}_5)\rtimes C_2
% in TeX

G:=Group("(C8xDic5):C2");
// GroupNames label

G:=SmallGroup(320,395);
// by ID

G=gap.SmallGroup(320,395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,100,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^10=d^2=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^4*b^5*c>;
// generators/relations

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