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G = C10×C22⋊C8order 320 = 26·5

Direct product of C10 and C22⋊C8

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

Aliases: C10×C22⋊C8, C232C40, C24.3C20, (C22×C8)⋊1C10, (C22×C10)⋊6C8, C222(C2×C40), (C22×C40)⋊5C2, C4.69(D4×C10), (C2×C40)⋊42C22, (C2×C20).535D4, C20.474(C2×D4), C2.1(C22×C40), (C23×C20).8C2, (C23×C4).5C10, (C23×C10).11C4, (C22×C20).47C4, C10.54(C22×C8), (C22×C4).11C20, C23.28(C2×C20), C2.3(C10×M4(2)), (C2×C20).981C23, C10.81(C2×M4(2)), (C2×C10).48M4(2), C20.161(C22⋊C4), C22.9(C5×M4(2)), C22.19(C22×C20), (C22×C20).495C22, (C2×C8)⋊10(C2×C10), (C2×C10)⋊13(C2×C8), (C2×C4).59(C2×C20), (C2×C4).145(C5×D4), C2.3(C10×C22⋊C4), C4.31(C5×C22⋊C4), (C2×C20).461(C2×C4), C10.138(C2×C22⋊C4), C22.32(C5×C22⋊C4), (C2×C4).149(C22×C10), (C2×C10).332(C22×C4), (C22×C10).182(C2×C4), (C22×C4).135(C2×C10), (C2×C10).198(C22⋊C4), SmallGroup(320,907)

Series: Derived Chief Lower central Upper central

C1C2 — C10×C22⋊C8
C1C2C4C2×C4C2×C20C2×C40C5×C22⋊C8 — C10×C22⋊C8
C1C2 — C10×C22⋊C8
C1C22×C20 — C10×C22⋊C8

Generators and relations for C10×C22⋊C8
 G = < a,b,c,d | a10=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 290 in 202 conjugacy classes, 114 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×2], C22, C22 [×10], C22 [×12], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C20 [×4], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C22⋊C8 [×4], C22×C8 [×2], C23×C4, C40 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊C8, C2×C40 [×4], C2×C40 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C5×C22⋊C8 [×4], C22×C40 [×2], C23×C20, C10×C22⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C40 [×4], C2×C20 [×6], C5×D4 [×4], C22×C10, C2×C22⋊C8, C5×C22⋊C4 [×4], C2×C40 [×6], C5×M4(2) [×2], C22×C20, D4×C10 [×2], C5×C22⋊C8 [×4], C10×C22⋊C4, C22×C40, C10×M4(2), C10×C22⋊C8

Smallest permutation representation of C10×C22⋊C8
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)
(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 101)(9 102)(10 103)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 91)(90 92)(111 139)(112 140)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 148)(122 149)(123 150)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)
(1 11 125 56 111 71 94 26)(2 12 126 57 112 72 95 27)(3 13 127 58 113 73 96 28)(4 14 128 59 114 74 97 29)(5 15 129 60 115 75 98 30)(6 16 130 51 116 76 99 21)(7 17 121 52 117 77 100 22)(8 18 122 53 118 78 91 23)(9 19 123 54 119 79 92 24)(10 20 124 55 120 80 93 25)(31 131 47 84 159 106 62 144)(32 132 48 85 160 107 63 145)(33 133 49 86 151 108 64 146)(34 134 50 87 152 109 65 147)(35 135 41 88 153 110 66 148)(36 136 42 89 154 101 67 149)(37 137 43 90 155 102 68 150)(38 138 44 81 156 103 69 141)(39 139 45 82 157 104 70 142)(40 140 46 83 158 105 61 143)

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

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

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

200 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L5A5B5C5D8A···8P10A···10AB10AC···10AR20A···20AF20AG···20AV40A···40BL
order12···222224···4444455558···810···1010···1020···2020···2040···40
size11···122221···1222211112···21···12···21···12···22···2

200 irreducible representations

dim111111111111112222
type+++++
imageC1C2C2C2C4C4C5C8C10C10C10C20C20C40D4M4(2)C5×D4C5×M4(2)
kernelC10×C22⋊C8C5×C22⋊C8C22×C40C23×C20C22×C20C23×C10C2×C22⋊C8C22×C10C22⋊C8C22×C8C23×C4C22×C4C24C23C2×C20C2×C10C2×C4C22
# reps142162416168424864441616

Matrix representation of C10×C22⋊C8 in GL4(𝔽41) generated by

25000
04000
0010
0001
,
40000
04000
0010
002740
,
1000
0100
00400
00040
,
40000
0300
002739
003714
G:=sub<GL(4,GF(41))| [25,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,27,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,3,0,0,0,0,27,37,0,0,39,14] >;

C10×C22⋊C8 in GAP, Magma, Sage, TeX

C_{10}\times C_2^2\rtimes C_8
% in TeX

G:=Group("C10xC2^2:C8");
// GroupNames label

G:=SmallGroup(320,907);
// by ID

G=gap.SmallGroup(320,907);
# by ID

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

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

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