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G = C10×C4.10D4order 320 = 26·5

Direct product of C10 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10×C4.10D4, C4.49(D4×C10), (C2×Q8).6C20, C20.456(C2×D4), (C2×C20).516D4, (C22×C4).5C20, (Q8×C10).26C4, (C22×C20).39C4, C23.30(C2×C20), (C22×Q8).3C10, (C2×C20).607C23, M4(2).7(C2×C10), C22.9(C22×C20), C20.126(C22⋊C4), (C2×M4(2)).11C10, (C10×M4(2)).29C2, (Q8×C10).248C22, (C22×C20).408C22, (C5×M4(2)).41C22, (C2×C4).5(C2×C20), (C2×C4).22(C5×D4), (Q8×C2×C10).13C2, C4.11(C5×C22⋊C4), (C2×C20).191(C2×C4), C2.15(C10×C22⋊C4), (C2×C4).2(C22×C10), (C2×Q8).33(C2×C10), C10.144(C2×C22⋊C4), (C22×C4).27(C2×C10), C22.19(C5×C22⋊C4), (C2×C10).263(C22×C4), (C22×C10).184(C2×C4), (C2×C10).201(C22⋊C4), SmallGroup(320,913)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C4.10D4
C1C2C4C2×C4C2×C20C5×M4(2)C5×C4.10D4 — C10×C4.10D4
C1C2C22 — C10×C4.10D4
C1C2×C10C22×C20 — C10×C4.10D4

Generators and relations for C10×C4.10D4
 G = < a,b,c,d | a10=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 210 in 146 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C40 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×Q8 [×8], C22×C10, C2×C4.10D4, C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C5×C4.10D4 [×4], C10×M4(2) [×2], Q8×C2×C10, C10×C4.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C4.10D4 [×2], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×C4.10D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C5×C4.10D4 [×2], C10×C22⋊C4, C10×C4.10D4

Smallest permutation representation of C10×C4.10D4
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 59 46 62)(2 60 47 63)(3 51 48 64)(4 52 49 65)(5 53 50 66)(6 54 41 67)(7 55 42 68)(8 56 43 69)(9 57 44 70)(10 58 45 61)(11 32 28 160)(12 33 29 151)(13 34 30 152)(14 35 21 153)(15 36 22 154)(16 37 23 155)(17 38 24 156)(18 39 25 157)(19 40 26 158)(20 31 27 159)(71 98 104 86)(72 99 105 87)(73 100 106 88)(74 91 107 89)(75 92 108 90)(76 93 109 81)(77 94 110 82)(78 95 101 83)(79 96 102 84)(80 97 103 85)(111 138 144 126)(112 139 145 127)(113 140 146 128)(114 131 147 129)(115 132 148 130)(116 133 149 121)(117 134 150 122)(118 135 141 123)(119 136 142 124)(120 137 143 125)
(1 139 67 150 46 127 54 117)(2 140 68 141 47 128 55 118)(3 131 69 142 48 129 56 119)(4 132 70 143 49 130 57 120)(5 133 61 144 50 121 58 111)(6 134 62 145 41 122 59 112)(7 135 63 146 42 123 60 113)(8 136 64 147 43 124 51 114)(9 137 65 148 44 125 52 115)(10 138 66 149 45 126 53 116)(11 92 37 80 28 90 155 103)(12 93 38 71 29 81 156 104)(13 94 39 72 30 82 157 105)(14 95 40 73 21 83 158 106)(15 96 31 74 22 84 159 107)(16 97 32 75 23 85 160 108)(17 98 33 76 24 86 151 109)(18 99 34 77 25 87 152 110)(19 100 35 78 26 88 153 101)(20 91 36 79 27 89 154 102)
(1 13 62 152 46 30 59 34)(2 14 63 153 47 21 60 35)(3 15 64 154 48 22 51 36)(4 16 65 155 49 23 52 37)(5 17 66 156 50 24 53 38)(6 18 67 157 41 25 54 39)(7 19 68 158 42 26 55 40)(8 20 69 159 43 27 56 31)(9 11 70 160 44 28 57 32)(10 12 61 151 45 29 58 33)(71 138 86 111 104 126 98 144)(72 139 87 112 105 127 99 145)(73 140 88 113 106 128 100 146)(74 131 89 114 107 129 91 147)(75 132 90 115 108 130 92 148)(76 133 81 116 109 121 93 149)(77 134 82 117 110 122 94 150)(78 135 83 118 101 123 95 141)(79 136 84 119 102 124 96 142)(80 137 85 120 103 125 97 143)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B5C5D8A···8H10A···10L10M···10T20A···20P20Q···20AF40A···40AF
order1222224444444455558···810···1010···1020···2020···2040···40
size1111222222444411114···41···12···22···24···44···4

110 irreducible representations

dim1111111111112244
type+++++-
imageC1C2C2C2C4C4C5C10C10C10C20C20D4C5×D4C4.10D4C5×C4.10D4
kernelC10×C4.10D4C5×C4.10D4C10×M4(2)Q8×C2×C10C22×C20Q8×C10C2×C4.10D4C4.10D4C2×M4(2)C22×Q8C22×C4C2×Q8C2×C20C2×C4C10C2
# reps14214441684161641628

Matrix representation of C10×C4.10D4 in GL6(𝔽41)

4000000
0400000
0018000
0001800
0000180
0000018
,
4000000
0400000
000401831
0010313
00004040
000021
,
40390000
010000
002304018
0038007
00140013
00390018
,
3200000
990000
0028323837
0032192020
00232403
003540035

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,18,3,40,2,0,0,31,13,40,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,23,38,1,39,0,0,0,0,40,0,0,0,40,0,0,0,0,0,18,7,13,18],[32,9,0,0,0,0,0,9,0,0,0,0,0,0,28,32,23,35,0,0,32,19,24,40,0,0,38,20,0,0,0,0,37,20,3,35] >;

C10×C4.10D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4._{10}D_4
% in TeX

G:=Group("C10xC4.10D4");
// GroupNames label

G:=SmallGroup(320,913);
// by ID

G=gap.SmallGroup(320,913);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,7004,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

׿
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