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## G = Dic5⋊C8order 160 = 25·5

### 3rd semidirect product of Dic5 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — Dic5⋊C8
 Lower central C5 — C10 — Dic5⋊C8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic5⋊C8
G = < a,b,c | a10=c8=1, b2=a5, bab-1=a-1, cac-1=a3, cbc-1=a5b >

Character table of Dic5⋊C8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 2 2 5 5 5 5 10 10 4 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -i i -i -i i i -i i 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 i -i i i -i -i i -i 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i -i i i i i -i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i i -i -i -i -i i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 1 -1 1 -1 i -i i i -i -i -1 1 1 ζ87 ζ83 ζ85 ζ8 ζ85 ζ8 ζ83 ζ87 -1 -1 1 -i i i -i linear of order 8 ρ10 1 -1 1 -1 i -i i i -i -i -1 1 1 ζ83 ζ87 ζ8 ζ85 ζ8 ζ85 ζ87 ζ83 -1 -1 1 -i i i -i linear of order 8 ρ11 1 -1 1 -1 i -i -i -i i i 1 -1 1 ζ8 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 ζ85 -1 -1 1 -i i i -i linear of order 8 ρ12 1 -1 1 -1 i -i -i -i i i 1 -1 1 ζ85 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 ζ8 -1 -1 1 -i i i -i linear of order 8 ρ13 1 -1 1 -1 -i i i i -i -i 1 -1 1 ζ83 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 ζ87 -1 -1 1 i -i -i i linear of order 8 ρ14 1 -1 1 -1 -i i i i -i -i 1 -1 1 ζ87 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 ζ83 -1 -1 1 i -i -i i linear of order 8 ρ15 1 -1 1 -1 -i i -i -i i i -1 1 1 ζ8 ζ85 ζ83 ζ87 ζ83 ζ87 ζ85 ζ8 -1 -1 1 i -i -i i linear of order 8 ρ16 1 -1 1 -1 -i i -i -i i i -1 1 1 ζ85 ζ8 ζ87 ζ83 ζ87 ζ83 ζ8 ζ85 -1 -1 1 i -i -i i linear of order 8 ρ17 2 2 -2 -2 0 0 2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 0 0 -2 2 -2 2 0 0 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 -2 2 0 0 2i -2i -2i 2i 0 0 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 -2 2 0 0 -2i 2i 2i -2i 0 0 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ21 4 4 4 4 -4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 4 4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 √5 √5 -√5 -√5 symplectic lifted from C22.F5, Schur index 2 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 -√5 -√5 √5 √5 symplectic lifted from C22.F5, Schur index 2 ρ25 4 -4 4 -4 4i -4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 i -i -i i complex lifted from D5⋊C8, Schur index 2 ρ26 4 -4 4 -4 -4i 4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -i i i -i complex lifted from D5⋊C8, Schur index 2 ρ27 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 √-5 -√-5 √-5 -√-5 complex lifted from C4⋊F5 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -√-5 √-5 -√-5 √-5 complex lifted from C4⋊F5

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

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

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

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

Matrix representation of Dic5⋊C8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 25 0 0 0 0 0 0 23 0 0 0 0 0 0 31 0 0 0 0 0 0 4
,
 17 4 0 0 0 0 30 24 0 0 0 0 0 0 0 32 0 0 0 0 32 0 0 0 0 0 0 0 0 9 0 0 0 0 9 0
,
 33 15 0 0 0 0 17 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,0,0,0,0,0,0,23,0,0,0,0,0,0,31,0,0,0,0,0,0,4],[17,30,0,0,0,0,4,24,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0],[33,17,0,0,0,0,15,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

Dic5⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes C_8
% in TeX

G:=Group("Dic5:C8");
// GroupNames label

G:=SmallGroup(160,79);
// by ID

G=gap.SmallGroup(160,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,86,2309,1169]);
// Polycyclic

G:=Group<a,b,c|a^10=c^8=1,b^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^5*b>;
// generators/relations

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