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G = C10×C8.C4order 320 = 26·5

Direct product of C10 and C8.C4

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

C1C4 — C10×C8.C4
C1C2C4C2×C4C2×C20C5×M4(2)C5×C8.C4 — C10×C8.C4
C1C2C4 — C10×C8.C4
C1C2×C20C22×C20 — C10×C8.C4

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 [×2], C2 [×2], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C20 [×4], C2×C10 [×3], C2×C10 [×2], C8.C4 [×4], C22×C8, C2×M4(2) [×2], C40 [×4], C40 [×4], C2×C20 [×6], C22×C10, C2×C8.C4, C2×C40 [×2], C2×C40 [×4], C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, C5×C8.C4 [×4], C22×C40, C10×M4(2) [×2], C10×C8.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C8.C4 [×2], C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, C2×C8.C4, C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C5×C8.C4 [×2], C10×C4⋊C4, C10×C8.C4

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

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B5C5D8A···8H8I···8P10A···10L10M···10T20A···20P20Q···20X40A···40AF40AG···40BL
order12222244444455558···88···810···1010···1020···2020···2040···4040···40
size11112211112211112···24···41···12···21···12···22···24···4

140 irreducible representations

dim111111111122222222
type+++++--
imageC1C2C2C2C4C5C10C10C10C20D4Q8Q8C8.C4C5×D4C5×Q8C5×Q8C5×C8.C4
kernelC10×C8.C4C5×C8.C4C22×C40C10×M4(2)C2×C40C2×C8.C4C8.C4C22×C8C2×M4(2)C2×C8C2×C20C2×C20C22×C10C10C2×C4C2×C4C23C2
# reps141284164832211884432

Matrix representation of C10×C8.C4 in GL3(𝔽41) generated by

2300
0310
0031
,
4000
0270
0038
,
100
001
0320
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

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