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G = C20⋊C8order 160 = 25·5

1st semidirect product of C20 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C201C8, Dic5.4Q8, Dic5.12D4, C10.1M4(2), C4⋊(C5⋊C8), C51(C4⋊C8), (C2×C4).6F5, C10.8(C2×C8), (C2×C20).5C4, C2.1(C4⋊F5), C10.5(C4⋊C4), C2.1(C4.F5), C22.9(C2×F5), (C2×Dic5).9C4, (C4×Dic5).12C2, (C2×Dic5).49C22, C2.4(C2×C5⋊C8), (C2×C5⋊C8).1C2, (C2×C10).4(C2×C4), SmallGroup(160,76)

Series: Derived Chief Lower central Upper central

C1C10 — C20⋊C8
C1C5C10Dic5C2×Dic5C2×C5⋊C8 — C20⋊C8
C5C10 — C20⋊C8
C1C22C2×C4

Generators and relations for C20⋊C8
 G = < a,b | a20=b8=1, bab-1=a3 >

5C4
5C4
10C4
5C2×C4
5C2×C4
10C8
10C8
2Dic5
5C42
5C2×C8
5C2×C8
2C5⋊C8
2C5⋊C8
5C4⋊C8

Character table of C20⋊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
ρ91-11-1-11ii-i-i-ii1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83-1-111-1-11    linear of order 8
ρ101-11-1-11ii-i-i-ii1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87-1-111-1-11    linear of order 8
ρ111-11-11-1-i-iii-ii1ζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85-1-11-111-1    linear of order 8
ρ121-11-11-1-i-iii-ii1ζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8-1-11-111-1    linear of order 8
ρ131-11-11-1ii-i-ii-i1ζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83-1-11-111-1    linear of order 8
ρ141-11-11-1ii-i-ii-i1ζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87-1-11-111-1    linear of order 8
ρ151-11-1-11-i-iiii-i1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8-1-111-1-11    linear of order 8
ρ161-11-1-11-i-iiii-i1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85-1-111-1-11    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
ρ234-44-44-4000000-10000000011-11-1-11    symplectic lifted from C5⋊C8, Schur index 2
ρ244-44-4-44000000-10000000011-1-111-1    symplectic lifted from C5⋊C8, Schur index 2
ρ2544-4-400000000-100000000-111-5--5-5--5    complex lifted from C4⋊F5
ρ2644-4-400000000-100000000-111--5-5--5-5    complex lifted from C4⋊F5
ρ274-4-4400000000-1000000001-11-5-5--5--5    complex lifted from C4.F5
ρ284-4-4400000000-1000000001-11--5--5-5-5    complex lifted from C4.F5

Smallest permutation representation of C20⋊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 58 113 152 31 100 69 129)(2 45 102 155 32 87 78 132)(3 52 111 158 33 94 67 135)(4 59 120 141 34 81 76 138)(5 46 109 144 35 88 65 121)(6 53 118 147 36 95 74 124)(7 60 107 150 37 82 63 127)(8 47 116 153 38 89 72 130)(9 54 105 156 39 96 61 133)(10 41 114 159 40 83 70 136)(11 48 103 142 21 90 79 139)(12 55 112 145 22 97 68 122)(13 42 101 148 23 84 77 125)(14 49 110 151 24 91 66 128)(15 56 119 154 25 98 75 131)(16 43 108 157 26 85 64 134)(17 50 117 160 27 92 73 137)(18 57 106 143 28 99 62 140)(19 44 115 146 29 86 71 123)(20 51 104 149 30 93 80 126)

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

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

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

C20⋊C8 is a maximal subgroup of
D20⋊C8  Dic101C8  C402C8  C401C8  C20.26M4(2)  Dic5.13D8  Dic5.23D8  Dic5.12Q16  C42.11F5  C42.12F5  C203M4(2)  C42.15F5  C5⋊C8⋊D4  D10⋊M4(2)  C20⋊C8⋊C2  C23.(C2×F5)  D202C8  Dic10⋊C8  C20⋊M4(2)  C4⋊C4.7F5  Dic5.M4(2)  C4⋊C4.9F5  C20.M4(2)  Dic5.12M4(2)  C208M4(2)  C20.30M4(2)  D4×C5⋊C8  C202M4(2)  Q8×C5⋊C8  C20.6M4(2)  C30.4M4(2)  C60⋊C8
C20⋊C8 is a maximal quotient of
C20⋊C16  C402C8  C401C8  C40.1C8  C10.(C4⋊C8)  C30.4M4(2)  C60⋊C8

Matrix representation of C20⋊C8 in GL6(𝔽41)

40390000
110000
00734347
003402727
00147140
00014714
,
28380000
32130000
00310818
00818031
0023132331
0010182810

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,7,34,14,0,0,0,34,0,7,14,0,0,34,27,14,7,0,0,7,27,0,14],[28,32,0,0,0,0,38,13,0,0,0,0,0,0,31,8,23,10,0,0,0,18,13,18,0,0,8,0,23,28,0,0,18,31,31,10] >;

C20⋊C8 in GAP, Magma, Sage, TeX

C_{20}\rtimes C_8
% in TeX

G:=Group("C20:C8");
// GroupNames label

G:=SmallGroup(160,76);
// by ID

G=gap.SmallGroup(160,76);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,2309,1169]);
// Polycyclic

G:=Group<a,b|a^20=b^8=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C20⋊C8 in TeX
Character table of C20⋊C8 in TeX

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