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## G = Q8⋊Dic5order 160 = 25·5

### 1st semidirect product of Q8 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q8⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — Q8⋊Dic5
 Lower central C5 — C10 — C20 — Q8⋊Dic5
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Q8⋊Dic5
G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Smallest permutation representation of Q8⋊Dic5
Regular action on 160 points
Generators in S160
(1 48 39 14)(2 49 40 15)(3 50 31 16)(4 41 32 17)(5 42 33 18)(6 43 34 19)(7 44 35 20)(8 45 36 11)(9 46 37 12)(10 47 38 13)(21 156 145 136)(22 157 146 137)(23 158 147 138)(24 159 148 139)(25 160 149 140)(26 151 150 131)(27 152 141 132)(28 153 142 133)(29 154 143 134)(30 155 144 135)(51 67 76 82)(52 68 77 83)(53 69 78 84)(54 70 79 85)(55 61 80 86)(56 62 71 87)(57 63 72 88)(58 64 73 89)(59 65 74 90)(60 66 75 81)(91 110 111 127)(92 101 112 128)(93 102 113 129)(94 103 114 130)(95 104 115 121)(96 105 116 122)(97 106 117 123)(98 107 118 124)(99 108 119 125)(100 109 120 126)
(1 74 39 59)(2 75 40 60)(3 76 31 51)(4 77 32 52)(5 78 33 53)(6 79 34 54)(7 80 35 55)(8 71 36 56)(9 72 37 57)(10 73 38 58)(11 87 45 62)(12 88 46 63)(13 89 47 64)(14 90 48 65)(15 81 49 66)(16 82 50 67)(17 83 41 68)(18 84 42 69)(19 85 43 70)(20 86 44 61)(21 111 145 91)(22 112 146 92)(23 113 147 93)(24 114 148 94)(25 115 149 95)(26 116 150 96)(27 117 141 97)(28 118 142 98)(29 119 143 99)(30 120 144 100)(101 137 128 157)(102 138 129 158)(103 139 130 159)(104 140 121 160)(105 131 122 151)(106 132 123 152)(107 133 124 153)(108 134 125 154)(109 135 126 155)(110 136 127 156)
(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 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 2 2 10 10 10 10 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 Q16 D10 Dic5 C5⋊D4 C5⋊D4 Q8⋊D5 C5⋊Q16 kernel Q8⋊Dic5 C2×C5⋊2C8 C4⋊Dic5 Q8×C10 C5×Q8 C20 C2×C10 C2×Q8 C10 C10 C2×C4 Q8 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2

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

 1 0 0 0 0 1 0 0 0 0 1 39 0 0 1 40
,
 1 0 0 0 0 1 0 0 0 0 21 13 0 0 7 20
,
 1 40 0 0 36 6 0 0 0 0 40 0 0 0 0 40
,
 15 14 0 0 19 26 0 0 0 0 33 4 0 0 35 8
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,21,7,0,0,13,20],[1,36,0,0,40,6,0,0,0,0,40,0,0,0,0,40],[15,19,0,0,14,26,0,0,0,0,33,35,0,0,4,8] >;

Q8⋊Dic5 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(160,42);
// by ID

G=gap.SmallGroup(160,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,579,297,69,4613]);
// Polycyclic

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

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