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## G = Dic5⋊Q8order 160 = 25·5

### 2nd semidirect product of Dic5 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊Q8
 Lower central C5 — C2×C10 — Dic5⋊Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Dic5⋊Q8
G = < a,b,c,d | a10=c4=1, b2=a5, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 176 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4×Dic5, C10.D4, C2×Dic10, Q8×C10, Dic5⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C5⋊D4, C22×D5, Q8×D5, C2×C5⋊D4, Dic5⋊Q8

Smallest permutation representation of Dic5⋊Q8
Regular action 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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(21 145 26 150)(22 144 27 149)(23 143 28 148)(24 142 29 147)(25 141 30 146)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 99 56 94)(52 98 57 93)(53 97 58 92)(54 96 59 91)(55 95 60 100)(61 104 66 109)(62 103 67 108)(63 102 68 107)(64 101 69 106)(65 110 70 105)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)(81 121 86 126)(82 130 87 125)(83 129 88 124)(84 128 89 123)(85 127 90 122)
(1 39 24 43)(2 40 25 44)(3 31 26 45)(4 32 27 46)(5 33 28 47)(6 34 29 48)(7 35 30 49)(8 36 21 50)(9 37 22 41)(10 38 23 42)(11 147 151 136)(12 148 152 137)(13 149 153 138)(14 150 154 139)(15 141 155 140)(16 142 156 131)(17 143 157 132)(18 144 158 133)(19 145 159 134)(20 146 160 135)(51 71 65 82)(52 72 66 83)(53 73 67 84)(54 74 68 85)(55 75 69 86)(56 76 70 87)(57 77 61 88)(58 78 62 89)(59 79 63 90)(60 80 64 81)(91 116 102 127)(92 117 103 128)(93 118 104 129)(94 119 105 130)(95 120 106 121)(96 111 107 122)(97 112 108 123)(98 113 109 124)(99 114 110 125)(100 115 101 126)
(1 59 24 63)(2 60 25 64)(3 51 26 65)(4 52 27 66)(5 53 28 67)(6 54 29 68)(7 55 30 69)(8 56 21 70)(9 57 22 61)(10 58 23 62)(11 122 151 111)(12 123 152 112)(13 124 153 113)(14 125 154 114)(15 126 155 115)(16 127 156 116)(17 128 157 117)(18 129 158 118)(19 130 159 119)(20 121 160 120)(31 82 45 71)(32 83 46 72)(33 84 47 73)(34 85 48 74)(35 86 49 75)(36 87 50 76)(37 88 41 77)(38 89 42 78)(39 90 43 79)(40 81 44 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D4 D5 D10 C5⋊D4 Q8×D5 kernel Dic5⋊Q8 C4×Dic5 C10.D4 C2×Dic10 Q8×C10 Dic5 C20 C2×Q8 C2×C4 C4 C2 # reps 1 1 4 1 1 4 2 2 6 8 4

Matrix representation of Dic5⋊Q8 in GL4(𝔽41) generated by

 34 40 0 0 8 1 0 0 0 0 40 0 0 0 0 40
,
 12 39 0 0 11 29 0 0 0 0 9 3 0 0 0 32
,
 17 35 0 0 7 24 0 0 0 0 34 32 0 0 1 7
,
 1 0 0 0 0 1 0 0 0 0 32 38 0 0 0 9
G:=sub<GL(4,GF(41))| [34,8,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[12,11,0,0,39,29,0,0,0,0,9,0,0,0,3,32],[17,7,0,0,35,24,0,0,0,0,34,1,0,0,32,7],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,38,9] >;

Dic5⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes Q_8
% in TeX

G:=Group("Dic5:Q8");
// GroupNames label

G:=SmallGroup(160,165);
// by ID

G=gap.SmallGroup(160,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,362,116,50,4613]);
// Polycyclic

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

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