direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C8.C4, C8.17(C2×C20), (C2×C40).53C4, (C2×C8).11C20, C4.77(D4×C10), (C2×C20).79Q8, C40.126(C2×C4), (C2×C20).538D4, C20.482(C2×D4), C23.9(C5×Q8), C20.102(C4⋊C4), C22.1(Q8×C10), (C22×C40).30C2, C4.28(C22×C20), (C22×C8).12C10, (C22×C10).21Q8, (C2×C20).902C23, (C2×C40).432C22, C20.245(C22×C4), M4(2).9(C2×C10), (C10×M4(2)).33C2, (C2×M4(2)).15C10, (C22×C20).590C22, (C5×M4(2)).43C22, C4.22(C5×C4⋊C4), C10.94(C2×C4⋊C4), C2.15(C10×C4⋊C4), (C2×C4).74(C5×D4), (C2×C4).21(C5×Q8), (C2×C8).90(C2×C10), (C2×C4).77(C2×C20), C22.11(C5×C4⋊C4), (C2×C10).14(C2×Q8), (C2×C10).93(C4⋊C4), (C2×C20).511(C2×C4), (C2×C4).77(C22×C10), (C22×C4).119(C2×C10), SmallGroup(320,930)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C8.C4
G = < a,b,c | a10=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 130 in 106 conjugacy classes, 82 normal (42 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, C23, C10, C10, C10, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C20, C2×C10, C2×C10, C8.C4, C22×C8, C2×M4(2), C40, C40, C2×C20, C22×C10, C2×C8.C4, C2×C40, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C5×C8.C4, C22×C40, C10×M4(2), C10×C8.C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C8.C4, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C8.C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C8.C4, C10×C4⋊C4, C10×C8.C4
►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 82 47 101 70 72 53 93)(2 83 48 102 61 73 54 94)(3 84 49 103 62 74 55 95)(4 85 50 104 63 75 56 96)(5 86 41 105 64 76 57 97)(6 87 42 106 65 77 58 98)(7 88 43 107 66 78 59 99)(8 89 44 108 67 79 60 100)(9 90 45 109 68 80 51 91)(10 81 46 110 69 71 52 92)(11 149 25 130 36 131 160 120)(12 150 26 121 37 132 151 111)(13 141 27 122 38 133 152 112)(14 142 28 123 39 134 153 113)(15 143 29 124 40 135 154 114)(16 144 30 125 31 136 155 115)(17 145 21 126 32 137 156 116)(18 146 22 127 33 138 157 117)(19 147 23 128 34 139 158 118)(20 148 24 129 35 140 159 119)
(1 141 58 127 70 133 42 117)(2 142 59 128 61 134 43 118)(3 143 60 129 62 135 44 119)(4 144 51 130 63 136 45 120)(5 145 52 121 64 137 46 111)(6 146 53 122 65 138 47 112)(7 147 54 123 66 139 48 113)(8 148 55 124 67 140 49 114)(9 149 56 125 68 131 50 115)(10 150 57 126 69 132 41 116)(11 96 30 80 36 104 155 90)(12 97 21 71 37 105 156 81)(13 98 22 72 38 106 157 82)(14 99 23 73 39 107 158 83)(15 100 24 74 40 108 159 84)(16 91 25 75 31 109 160 85)(17 92 26 76 32 110 151 86)(18 93 27 77 33 101 152 87)(19 94 28 78 34 102 153 88)(20 95 29 79 35 103 154 89)
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), (1,82,47,101,70,72,53,93)(2,83,48,102,61,73,54,94)(3,84,49,103,62,74,55,95)(4,85,50,104,63,75,56,96)(5,86,41,105,64,76,57,97)(6,87,42,106,65,77,58,98)(7,88,43,107,66,78,59,99)(8,89,44,108,67,79,60,100)(9,90,45,109,68,80,51,91)(10,81,46,110,69,71,52,92)(11,149,25,130,36,131,160,120)(12,150,26,121,37,132,151,111)(13,141,27,122,38,133,152,112)(14,142,28,123,39,134,153,113)(15,143,29,124,40,135,154,114)(16,144,30,125,31,136,155,115)(17,145,21,126,32,137,156,116)(18,146,22,127,33,138,157,117)(19,147,23,128,34,139,158,118)(20,148,24,129,35,140,159,119), (1,141,58,127,70,133,42,117)(2,142,59,128,61,134,43,118)(3,143,60,129,62,135,44,119)(4,144,51,130,63,136,45,120)(5,145,52,121,64,137,46,111)(6,146,53,122,65,138,47,112)(7,147,54,123,66,139,48,113)(8,148,55,124,67,140,49,114)(9,149,56,125,68,131,50,115)(10,150,57,126,69,132,41,116)(11,96,30,80,36,104,155,90)(12,97,21,71,37,105,156,81)(13,98,22,72,38,106,157,82)(14,99,23,73,39,107,158,83)(15,100,24,74,40,108,159,84)(16,91,25,75,31,109,160,85)(17,92,26,76,32,110,151,86)(18,93,27,77,33,101,152,87)(19,94,28,78,34,102,153,88)(20,95,29,79,35,103,154,89)>;
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), (1,82,47,101,70,72,53,93)(2,83,48,102,61,73,54,94)(3,84,49,103,62,74,55,95)(4,85,50,104,63,75,56,96)(5,86,41,105,64,76,57,97)(6,87,42,106,65,77,58,98)(7,88,43,107,66,78,59,99)(8,89,44,108,67,79,60,100)(9,90,45,109,68,80,51,91)(10,81,46,110,69,71,52,92)(11,149,25,130,36,131,160,120)(12,150,26,121,37,132,151,111)(13,141,27,122,38,133,152,112)(14,142,28,123,39,134,153,113)(15,143,29,124,40,135,154,114)(16,144,30,125,31,136,155,115)(17,145,21,126,32,137,156,116)(18,146,22,127,33,138,157,117)(19,147,23,128,34,139,158,118)(20,148,24,129,35,140,159,119), (1,141,58,127,70,133,42,117)(2,142,59,128,61,134,43,118)(3,143,60,129,62,135,44,119)(4,144,51,130,63,136,45,120)(5,145,52,121,64,137,46,111)(6,146,53,122,65,138,47,112)(7,147,54,123,66,139,48,113)(8,148,55,124,67,140,49,114)(9,149,56,125,68,131,50,115)(10,150,57,126,69,132,41,116)(11,96,30,80,36,104,155,90)(12,97,21,71,37,105,156,81)(13,98,22,72,38,106,157,82)(14,99,23,73,39,107,158,83)(15,100,24,74,40,108,159,84)(16,91,25,75,31,109,160,85)(17,92,26,76,32,110,151,86)(18,93,27,77,33,101,152,87)(19,94,28,78,34,102,153,88)(20,95,29,79,35,103,154,89) );
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)], [(1,82,47,101,70,72,53,93),(2,83,48,102,61,73,54,94),(3,84,49,103,62,74,55,95),(4,85,50,104,63,75,56,96),(5,86,41,105,64,76,57,97),(6,87,42,106,65,77,58,98),(7,88,43,107,66,78,59,99),(8,89,44,108,67,79,60,100),(9,90,45,109,68,80,51,91),(10,81,46,110,69,71,52,92),(11,149,25,130,36,131,160,120),(12,150,26,121,37,132,151,111),(13,141,27,122,38,133,152,112),(14,142,28,123,39,134,153,113),(15,143,29,124,40,135,154,114),(16,144,30,125,31,136,155,115),(17,145,21,126,32,137,156,116),(18,146,22,127,33,138,157,117),(19,147,23,128,34,139,158,118),(20,148,24,129,35,140,159,119)], [(1,141,58,127,70,133,42,117),(2,142,59,128,61,134,43,118),(3,143,60,129,62,135,44,119),(4,144,51,130,63,136,45,120),(5,145,52,121,64,137,46,111),(6,146,53,122,65,138,47,112),(7,147,54,123,66,139,48,113),(8,148,55,124,67,140,49,114),(9,149,56,125,68,131,50,115),(10,150,57,126,69,132,41,116),(11,96,30,80,36,104,155,90),(12,97,21,71,37,105,156,81),(13,98,22,72,38,106,157,82),(14,99,23,73,39,107,158,83),(15,100,24,74,40,108,159,84),(16,91,25,75,31,109,160,85),(17,92,26,76,32,110,151,86),(18,93,27,77,33,101,152,87),(19,94,28,78,34,102,153,88),(20,95,29,79,35,103,154,89)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF | 40AG | ··· | 40BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | Q8 | Q8 | C8.C4 | C5×D4 | C5×Q8 | C5×Q8 | C5×C8.C4 |
| kernel | C10×C8.C4 | C5×C8.C4 | C22×C40 | C10×M4(2) | C2×C40 | C2×C8.C4 | C8.C4 | C22×C8 | C2×M4(2) | C2×C8 | C2×C20 | C2×C20 | C22×C10 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 2 | 8 | 4 | 16 | 4 | 8 | 32 | 2 | 1 | 1 | 8 | 8 | 4 | 4 | 32 |
Matrix representation of C10×C8.C4 ►in GL3(𝔽41) generated by
| 23 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 31 |
| 40 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 38 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 32 | 0 |
G:=sub<GL(3,GF(41))| [23,0,0,0,31,0,0,0,31],[40,0,0,0,27,0,0,0,38],[1,0,0,0,0,32,0,1,0] >;
C10×C8.C4 in GAP, Magma, Sage, TeX
C_{10}\times C_8.C_4 % in TeX
G:=Group("C10xC8.C4"); // GroupNames label
G:=SmallGroup(320,930);
// by ID
G=gap.SmallGroup(320,930);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,7004,172,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations