metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊1C8, Dic5.4Q8, Dic5.12D4, C10.1M4(2), C4⋊(C5⋊C8), C5⋊1(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
Generators and relations for C20⋊C8
G = < a,b | a20=b8=1, bab-1=a3 >
Character table of C20⋊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 | -1 | 1 | i | i | -i | -i | -i | i | 1 | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | i | 1 | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | 1 | ζ87 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | ζ85 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | 1 | ζ83 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | ζ8 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | 1 | ζ8 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | ζ83 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | 1 | ζ85 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | ζ87 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | 1 | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | 1 | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 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 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ24 | 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 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ25 | 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 |
| ρ26 | 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 |
| 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 |
►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 53 33 142 139 76 112 88)(2 60 22 145 140 63 101 91)(3 47 31 148 121 70 110 94)(4 54 40 151 122 77 119 97)(5 41 29 154 123 64 108 100)(6 48 38 157 124 71 117 83)(7 55 27 160 125 78 106 86)(8 42 36 143 126 65 115 89)(9 49 25 146 127 72 104 92)(10 56 34 149 128 79 113 95)(11 43 23 152 129 66 102 98)(12 50 32 155 130 73 111 81)(13 57 21 158 131 80 120 84)(14 44 30 141 132 67 109 87)(15 51 39 144 133 74 118 90)(16 58 28 147 134 61 107 93)(17 45 37 150 135 68 116 96)(18 52 26 153 136 75 105 99)(19 59 35 156 137 62 114 82)(20 46 24 159 138 69 103 85)
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), (1,53,33,142,139,76,112,88)(2,60,22,145,140,63,101,91)(3,47,31,148,121,70,110,94)(4,54,40,151,122,77,119,97)(5,41,29,154,123,64,108,100)(6,48,38,157,124,71,117,83)(7,55,27,160,125,78,106,86)(8,42,36,143,126,65,115,89)(9,49,25,146,127,72,104,92)(10,56,34,149,128,79,113,95)(11,43,23,152,129,66,102,98)(12,50,32,155,130,73,111,81)(13,57,21,158,131,80,120,84)(14,44,30,141,132,67,109,87)(15,51,39,144,133,74,118,90)(16,58,28,147,134,61,107,93)(17,45,37,150,135,68,116,96)(18,52,26,153,136,75,105,99)(19,59,35,156,137,62,114,82)(20,46,24,159,138,69,103,85)>;
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), (1,53,33,142,139,76,112,88)(2,60,22,145,140,63,101,91)(3,47,31,148,121,70,110,94)(4,54,40,151,122,77,119,97)(5,41,29,154,123,64,108,100)(6,48,38,157,124,71,117,83)(7,55,27,160,125,78,106,86)(8,42,36,143,126,65,115,89)(9,49,25,146,127,72,104,92)(10,56,34,149,128,79,113,95)(11,43,23,152,129,66,102,98)(12,50,32,155,130,73,111,81)(13,57,21,158,131,80,120,84)(14,44,30,141,132,67,109,87)(15,51,39,144,133,74,118,90)(16,58,28,147,134,61,107,93)(17,45,37,150,135,68,116,96)(18,52,26,153,136,75,105,99)(19,59,35,156,137,62,114,82)(20,46,24,159,138,69,103,85) );
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)], [(1,53,33,142,139,76,112,88),(2,60,22,145,140,63,101,91),(3,47,31,148,121,70,110,94),(4,54,40,151,122,77,119,97),(5,41,29,154,123,64,108,100),(6,48,38,157,124,71,117,83),(7,55,27,160,125,78,106,86),(8,42,36,143,126,65,115,89),(9,49,25,146,127,72,104,92),(10,56,34,149,128,79,113,95),(11,43,23,152,129,66,102,98),(12,50,32,155,130,73,111,81),(13,57,21,158,131,80,120,84),(14,44,30,141,132,67,109,87),(15,51,39,144,133,74,118,90),(16,58,28,147,134,61,107,93),(17,45,37,150,135,68,116,96),(18,52,26,153,136,75,105,99),(19,59,35,156,137,62,114,82),(20,46,24,159,138,69,103,85)]])
C20⋊C8 is a maximal subgroup of
D20⋊C8 Dic10⋊1C8 C40⋊2C8 C40⋊1C8 C20.26M4(2) Dic5.13D8 Dic5.23D8 Dic5.12Q16 C42.11F5 C42.12F5 C20⋊3M4(2) C42.15F5 C5⋊C8⋊D4 D10⋊M4(2) C20⋊C8⋊C2 C23.(C2×F5) D20⋊2C8 Dic10⋊C8 C20⋊M4(2) C4⋊C4.7F5 Dic5.M4(2) C4⋊C4.9F5 C20.M4(2) Dic5.12M4(2) C20⋊8M4(2) C20.30M4(2) D4×C5⋊C8 C20⋊2M4(2) Q8×C5⋊C8 C20.6M4(2) C30.4M4(2) C60⋊C8
C20⋊C8 is a maximal quotient of
C20⋊C16 C40⋊2C8 C40⋊1C8 C40.1C8 C10.(C4⋊C8) C30.4M4(2) C60⋊C8
Matrix representation of C20⋊C8 ►in GL6(𝔽41)
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 34 | 7 |
| 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 14 | 7 | 14 |
| 28 | 38 | 0 | 0 | 0 | 0 |
| 32 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 8 | 18 |
| 0 | 0 | 8 | 18 | 0 | 31 |
| 0 | 0 | 23 | 13 | 23 | 31 |
| 0 | 0 | 10 | 18 | 28 | 10 |
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