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G = C10×D4⋊C4order 320 = 26·5

Direct product of C10 and D4⋊C4

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

Aliases: C10×D4⋊C4, D43(C2×C20), (C2×D4)⋊7C20, C2.1(C10×D8), (D4×C10)⋊31C4, (C22×C8)⋊3C10, (C22×C40)⋊7C2, C4.51(D4×C10), C10.73(C2×D8), (C2×C10).53D8, (C2×C40)⋊43C22, C20.458(C2×D4), (C2×C20).414D4, C4.1(C22×C20), C2.1(C10×SD16), C22.12(C5×D8), C23.54(C5×D4), (C2×C10).44SD16, C10.81(C2×SD16), (C22×D4).6C10, C22.41(D4×C10), (C2×C20).890C23, C20.205(C22×C4), (C22×C10).215D4, C22.10(C5×SD16), C20.127(C22⋊C4), (D4×C10).286C22, (C22×C20).581C22, (C2×C4⋊C4)⋊9C10, C4⋊C47(C2×C10), (C10×C4⋊C4)⋊36C2, (C2×C8)⋊11(C2×C10), (C5×D4)⋊33(C2×C4), (D4×C2×C10).18C2, (C2×C4).68(C5×D4), (C5×C4⋊C4)⋊63C22, (C2×C4).47(C2×C20), C4.12(C5×C22⋊C4), (C2×C20).441(C2×C4), (C2×D4).44(C2×C10), (C2×C10).617(C2×D4), C2.17(C10×C22⋊C4), C10.146(C2×C22⋊C4), (C2×C4).65(C22×C10), C22.33(C5×C22⋊C4), (C22×C4).110(C2×C10), (C2×C10).202(C22⋊C4), SmallGroup(320,915)

Series: Derived Chief Lower central Upper central

C1C4 — C10×D4⋊C4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×D4⋊C4 — C10×D4⋊C4
C1C2C4 — C10×D4⋊C4
C1C22×C10C22×C20 — C10×D4⋊C4

Generators and relations for C10×D4⋊C4
 G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 402 in 202 conjugacy classes, 98 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], C10 [×3], C10 [×4], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C10 [×16], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×C10, C22×C10 [×10], C2×D4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C2×C40 [×2], C22×C20, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, C5×D4⋊C4 [×4], C10×C4⋊C4, C22×C40, D4×C2×C10, C10×D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×D4⋊C4, C5×C22⋊C4 [×4], C5×D8 [×2], C5×SD16 [×2], C22×C20, D4×C10 [×2], C5×D4⋊C4 [×4], C10×C22⋊C4, C10×D8, C10×SD16, C10×D4⋊C4

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

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H5A5B5C5D8A···8H10A···10AB10AC···10AR20A···20P20Q···20AF40A···40AF
order12···222224444444455558···810···1010···1020···2020···2040···40
size11···144442222444411112···21···14···42···24···42···2

140 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4D8SD16C5×D4C5×D4C5×D8C5×SD16
kernelC10×D4⋊C4C5×D4⋊C4C10×C4⋊C4C22×C40D4×C2×C10D4×C10C2×D4⋊C4D4⋊C4C2×C4⋊C4C22×C8C22×D4C2×D4C2×C20C22×C10C2×C10C2×C10C2×C4C23C22C22
# reps1411184164443231441241616

Matrix representation of C10×D4⋊C4 in GL4(𝔽41) generated by

40000
02500
00250
00025
,
1000
0100
00139
00140
,
40000
0100
00139
00040
,
1000
03200
00017
00290
G:=sub<GL(4,GF(41))| [40,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[1,0,0,0,0,32,0,0,0,0,0,29,0,0,17,0] >;

C10×D4⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times D_4\rtimes C_4
% in TeX

G:=Group("C10xD4:C4");
// GroupNames label

G:=SmallGroup(320,915);
// by ID

G=gap.SmallGroup(320,915);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172]);
// Polycyclic

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

׿
×
𝔽