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G = Dic5⋊C8order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic53C8, Dic5.5Q8, Dic5.13D4, C10.4M4(2), C52(C4⋊C8), (C2×C4).4F5, (C2×C20).4C4, C10.4(C2×C8), C2.2(C4⋊F5), C10.6(C4⋊C4), C2.5(D5⋊C8), (C4×Dic5).8C2, C22.12(C2×F5), (C2×Dic5).11C4, C2.2(C22.F5), (C2×Dic5).52C22, (C2×C5⋊C8).3C2, (C2×C10).7(C2×C4), SmallGroup(160,79)

Series: Derived Chief Lower central Upper central

C1C10 — Dic5⋊C8
C1C5C10Dic5C2×Dic5C2×C5⋊C8 — Dic5⋊C8
C5C10 — Dic5⋊C8
C1C22C2×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 >

2C4
5C4
5C4
5C4
5C4
5C2×C4
5C2×C4
10C8
10C8
2C20
5C42
5C2×C8
5C2×C8
2C5⋊C8
2C5⋊C8
5C4⋊C8

Character table of Dic5⋊C8

 class 12A2B2C4A4B4C4D4E4F4G4H58A8B8C8D8E8F8G8H10A10B10C20A20B20C20D
 size 11112255551010410101010101010104444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-11-1111-1-1-11111-1-1-1-1    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ41111-1-11111-1-111-1-1-1111-1111-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-1-11-ii-i-iii-ii1111111    linear of order 4
ρ6111111-1-1-1-1-1-11i-iii-i-ii-i1111111    linear of order 4
ρ71111-1-1-1-1-1-1111-i-iiiii-i-i111-1-1-1-1    linear of order 4
ρ81111-1-1-1-1-1-1111ii-i-i-i-iii111-1-1-1-1    linear of order 4
ρ91-11-1i-iii-i-i-111ζ87ζ83ζ85ζ8ζ85ζ8ζ83ζ87-1-11-iii-i    linear of order 8
ρ101-11-1i-iii-i-i-111ζ83ζ87ζ8ζ85ζ8ζ85ζ87ζ83-1-11-iii-i    linear of order 8
ρ111-11-1i-i-i-iii1-11ζ8ζ8ζ87ζ83ζ83ζ87ζ85ζ85-1-11-iii-i    linear of order 8
ρ121-11-1i-i-i-iii1-11ζ85ζ85ζ83ζ87ζ87ζ83ζ8ζ8-1-11-iii-i    linear of order 8
ρ131-11-1-iiii-i-i1-11ζ83ζ83ζ85ζ8ζ8ζ85ζ87ζ87-1-11i-i-ii    linear of order 8
ρ141-11-1-iiii-i-i1-11ζ87ζ87ζ8ζ85ζ85ζ8ζ83ζ83-1-11i-i-ii    linear of order 8
ρ151-11-1-ii-i-iii-111ζ8ζ85ζ83ζ87ζ83ζ87ζ85ζ8-1-11i-i-ii    linear of order 8
ρ161-11-1-ii-i-iii-111ζ85ζ8ζ87ζ83ζ87ζ83ζ8ζ85-1-11i-i-ii    linear of order 8
ρ1722-2-2002-22-2002000000002-2-20000    orthogonal lifted from D4
ρ1822-2-200-22-22002000000002-2-20000    symplectic lifted from Q8, Schur index 2
ρ192-2-22002i-2i-2i2i00200000000-22-20000    complex lifted from M4(2)
ρ202-2-2200-2i2i2i-2i00200000000-22-20000    complex lifted from M4(2)
ρ214444-4-4000000-100000000-1-1-11111    orthogonal lifted from C2×F5
ρ22444444000000-100000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ234-4-4400000000-1000000001-1155-5-5    symplectic lifted from C22.F5, Schur index 2
ρ244-4-4400000000-1000000001-11-5-555    symplectic lifted from C22.F5, Schur index 2
ρ254-44-44i-4i000000-10000000011-1i-i-ii    complex lifted from D5⋊C8, Schur index 2
ρ264-44-4-4i4i000000-10000000011-1-iii-i    complex lifted from D5⋊C8, Schur index 2
ρ2744-4-400000000-100000000-111-5--5-5--5    complex lifted from C4⋊F5
ρ2844-4-400000000-100000000-111--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)]])

Dic5⋊C8 is a maximal subgroup of
Dic5.D8  Dic5.SD16  Dic5.Q16  C42.6F5  C42.11F5  C42.14F5  C42.7F5  C5⋊C88D4  Dic5⋊M4(2)  C20⋊C8⋊C2  C23.(C2×F5)  Dic10⋊C8  D102M4(2)  C4⋊C4.7F5  Dic5.M4(2)  C4⋊C4.9F5  C20.34M4(2)  Dic5.13M4(2)  C208M4(2)  C5⋊C87D4  C20.6M4(2)  Dic15⋊C8  Dic5.13D12
Dic5⋊C8 is a maximal quotient of
C42.9F5  C20.26M4(2)  Dic5.13D8  C10.M5(2)  C10.(C4⋊C8)  Dic15⋊C8  Dic5.13D12

Matrix representation of Dic5⋊C8 in GL6(𝔽41)

4000000
0400000
0025000
0002300
0000310
000004
,
1740000
30240000
0003200
0032000
000009
000090
,
33150000
1780000
000010
000001
000100
001000

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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Subgroup lattice of Dic5⋊C8 in TeX
Character table of Dic5⋊C8 in TeX

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