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G = C10×C422C2order 320 = 26·5

Direct product of C10 and C422C2

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

Aliases: C10×C422C2, (C2×C42)⋊4C10, (C4×C20)⋊49C22, C4214(C2×C10), C24.14(C2×C10), (C2×C20).712C23, (C2×C10).348C24, (C23×C10).14C22, C22.22(C23×C10), C23.72(C22×C10), (C22×C10).86C23, (C22×C20).510C22, (C2×C4×C20)⋊6C2, (C2×C4⋊C4)⋊17C10, (C10×C4⋊C4)⋊44C2, C4⋊C413(C2×C10), (C5×C4⋊C4)⋊69C22, C2.11(C10×C4○D4), C10.230(C2×C4○D4), C22.34(C5×C4○D4), (C10×C22⋊C4).32C2, C22⋊C4.11(C2×C10), (C2×C22⋊C4).12C10, (C2×C4).16(C22×C10), (C2×C10).234(C4○D4), (C22×C4).102(C2×C10), (C5×C22⋊C4).145C22, SmallGroup(320,1530)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C422C2
C1C2C22C2×C10C22×C10C5×C22⋊C4C5×C422C2 — C10×C422C2
C1C22 — C10×C422C2
C1C22×C10 — C10×C422C2

Generators and relations for C10×C422C2
 G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 354 in 246 conjugacy classes, 162 normal (12 characteristic)
C1, C2 [×7], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×12], C2×C4 [×12], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C24, C20 [×12], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C422C2 [×8], C2×C20 [×12], C2×C20 [×12], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C422C2, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×12], C22×C20 [×6], C23×C10, C2×C4×C20, C10×C22⋊C4 [×3], C10×C4⋊C4 [×3], C5×C422C2 [×8], C10×C422C2
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×6], C24, C2×C10 [×35], C422C2 [×4], C2×C4○D4 [×3], C22×C10 [×15], C2×C422C2, C5×C4○D4 [×6], C23×C10, C5×C422C2 [×4], C10×C4○D4 [×3], C10×C422C2

Smallest permutation representation of C10×C422C2
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)
(1 123 51 136)(2 124 52 137)(3 125 53 138)(4 126 54 139)(5 127 55 140)(6 128 56 131)(7 129 57 132)(8 130 58 133)(9 121 59 134)(10 122 60 135)(11 74 159 107)(12 75 160 108)(13 76 151 109)(14 77 152 110)(15 78 153 101)(16 79 154 102)(17 80 155 103)(18 71 156 104)(19 72 157 105)(20 73 158 106)(21 88 40 91)(22 89 31 92)(23 90 32 93)(24 81 33 94)(25 82 34 95)(26 83 35 96)(27 84 36 97)(28 85 37 98)(29 86 38 99)(30 87 39 100)(41 112 70 145)(42 113 61 146)(43 114 62 147)(44 115 63 148)(45 116 64 149)(46 117 65 150)(47 118 66 141)(48 119 67 142)(49 120 68 143)(50 111 69 144)
(1 79 67 96)(2 80 68 97)(3 71 69 98)(4 72 70 99)(5 73 61 100)(6 74 62 91)(7 75 63 92)(8 76 64 93)(9 77 65 94)(10 78 66 95)(11 114 40 131)(12 115 31 132)(13 116 32 133)(14 117 33 134)(15 118 34 135)(16 119 35 136)(17 120 36 137)(18 111 37 138)(19 112 38 139)(20 113 39 140)(21 128 159 147)(22 129 160 148)(23 130 151 149)(24 121 152 150)(25 122 153 141)(26 123 154 142)(27 124 155 143)(28 125 156 144)(29 126 157 145)(30 127 158 146)(41 86 54 105)(42 87 55 106)(43 88 56 107)(44 89 57 108)(45 90 58 109)(46 81 59 110)(47 82 60 101)(48 83 51 102)(49 84 52 103)(50 85 53 104)
(11 159)(12 160)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 138)(112 139)(113 140)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)(121 150)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R5A5B5C5D10A···10AB10AC···10AJ20A···20AV20AW···20BT
order12···2224···44···4555510···1010···1020···2020···20
size11···1442···24···411111···14···42···24···4

140 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C2C5C10C10C10C10C4○D4C5×C4○D4
kernelC10×C422C2C2×C4×C20C10×C22⋊C4C10×C4⋊C4C5×C422C2C2×C422C2C2×C42C2×C22⋊C4C2×C4⋊C4C422C2C2×C10C22
# reps11338441212321248

Matrix representation of C10×C422C2 in GL5(𝔽41)

400000
023000
002300
000180
000018
,
400000
0203900
0162100
000320
000032
,
400000
032000
003200
000040
000400
,
400000
01000
0204000
00010
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,18,0,0,0,0,0,18],[40,0,0,0,0,0,20,16,0,0,0,39,21,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,40,0],[40,0,0,0,0,0,1,20,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;

C10×C422C2 in GAP, Magma, Sage, TeX

C_{10}\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C10xC4^2:2C2");
// GroupNames label

G:=SmallGroup(320,1530);
// by ID

G=gap.SmallGroup(320,1530);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,436]);
// Polycyclic

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

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