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G = C8xDic5order 160 = 25·5

Direct product of C8 and Dic5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8xDic5, C40:7C4, C10.8C42, C5:4(C4xC8), C5:2C8:9C4, C2.2(C8xD5), (C2xC8).10D5, C4.20(C4xD5), C10.12(C2xC8), (C2xC40).11C2, C20.61(C2xC4), (C2xC4).90D10, C2.2(C4xDic5), C22.8(C4xD5), C4.12(C2xDic5), (C4xDic5).14C2, (C2xDic5).15C4, (C2xC20).104C22, (C2xC5:2C8).14C2, (C2xC10).29(C2xC4), SmallGroup(160,20)

Series: Derived Chief Lower central Upper central

C1C5 — C8xDic5
C1C5C10C2xC10C2xC20C4xDic5 — C8xDic5
C5 — C8xDic5
C1C2xC8

Generators and relations for C8xDic5
 G = < a,b,c | a8=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 88 in 44 conjugacy classes, 33 normal (17 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, D5, C42, C2xC8, Dic5, D10, C4xC8, C4xD5, C2xDic5, C8xD5, C4xDic5, C8xDic5
5C4
5C4
5C4
5C4
5C8
5C2xC4
5C2xC4
5C8
5C42
5C2xC8
5C4xC8

Smallest permutation representation of C8xDic5
Regular action on 160 points
Generators in S160
(1 90 48 74 39 63 14 55)(2 81 49 75 40 64 15 56)(3 82 50 76 31 65 16 57)(4 83 41 77 32 66 17 58)(5 84 42 78 33 67 18 59)(6 85 43 79 34 68 19 60)(7 86 44 80 35 69 20 51)(8 87 45 71 36 70 11 52)(9 88 46 72 37 61 12 53)(10 89 47 73 38 62 13 54)(21 127 156 111 131 96 144 109)(22 128 157 112 132 97 145 110)(23 129 158 113 133 98 146 101)(24 130 159 114 134 99 147 102)(25 121 160 115 135 100 148 103)(26 122 151 116 136 91 149 104)(27 123 152 117 137 92 150 105)(28 124 153 118 138 93 141 106)(29 125 154 119 139 94 142 107)(30 126 155 120 140 95 143 108)
(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)
(1 104 6 109)(2 103 7 108)(3 102 8 107)(4 101 9 106)(5 110 10 105)(11 94 16 99)(12 93 17 98)(13 92 18 97)(14 91 19 96)(15 100 20 95)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 143 56 148)(52 142 57 147)(53 141 58 146)(54 150 59 145)(55 149 60 144)(61 138 66 133)(62 137 67 132)(63 136 68 131)(64 135 69 140)(65 134 70 139)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

C8xDic5 is a maximal subgroup of
C40.88D4  C80:17C4  C40.9Q8  C40:C8  C20.31M4(2)  C40:2C8  C40:1C8  C20.26M4(2)  Dic5.13D8  Dic5:C16  C40.C8  C10.M5(2)  C40.1C8  D5xC4xC8  D10.5C42  C40:Q8  D10.6C42  D10.7C42  Dic5.14M4(2)  C40:8C4:C2  C5:5(C8xD4)  Dic5:2M4(2)  Dic5:4D8  Dic5:6SD16  Dic5.5D8  (C8xDic5):C2  Dic5:7SD16  Dic5:4Q16  Dic5.3Q16  Q8:Dic5:C2  Dic5.5M4(2)  Dic10:5C8  C42.200D10  C42.31D10  Dic5:8SD16  C40:5Q8  C8.8Dic10  D40:12C4  Dic5:5Q16  C40:2Q8  C8.6Dic10  D40:13C4  C20.42C42  C20.37C42  C40:18D4  C40.93D4  C40:5D4  C40.22D4  C40.43D4  C40:15D4  C40.26D4  C40.28D4  D8:5Dic5  Dic15:4C8
C8xDic5 is a maximal quotient of
C42.279D10  C80:17C4  (C2xC40):15C4  Dic15:4C8

64 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L5A5B8A···8H8I···8P10A···10F20A···20H40A···40P
order122244444···4558···88···810···1020···2040···40
size111111115···5221···15···52···22···22···2

64 irreducible representations

dim11111111222222
type+++++-+
imageC1C2C2C2C4C4C4C8D5Dic5D10C4xD5C4xD5C8xD5
kernelC8xDic5C2xC5:2C8C4xDic5C2xC40C5:2C8C40C2xDic5Dic5C2xC8C8C2xC4C4C22C2
# reps1111444162424416

Matrix representation of C8xDic5 in GL3(F41) generated by

100
0140
0014
,
4000
0040
0135
,
3200
0215
02739
G:=sub<GL(3,GF(41))| [1,0,0,0,14,0,0,0,14],[40,0,0,0,0,1,0,40,35],[32,0,0,0,2,27,0,15,39] >;

C8xDic5 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_5
% in TeX

G:=Group("C8xDic5");
// GroupNames label

G:=SmallGroup(160,20);
// by ID

G=gap.SmallGroup(160,20);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,55,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^8=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C8xDic5 in TeX

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