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G = C10×C41D4order 320 = 26·5

Direct product of C10 and C41D4

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

Aliases: C10×C41D4, C41(D4×C10), (C2×C20)⋊33D4, C2012(C2×D4), (C2×C42)⋊11C10, C4218(C2×C10), (C4×C20)⋊59C22, (C22×D4)⋊5C10, (D4×C10)⋊63C22, C24.15(C2×C10), C22.63(D4×C10), (C2×C20).961C23, (C2×C10).350C24, C10.186(C22×D4), C23.8(C22×C10), (C23×C10).15C22, C22.24(C23×C10), (C22×C10).88C23, (C22×C20).596C22, (C2×C4×C20)⋊24C2, (C2×C4)⋊7(C5×D4), (D4×C2×C10)⋊20C2, C2.10(D4×C2×C10), (C2×D4)⋊11(C2×C10), (C2×C10).684(C2×D4), (C2×C4).136(C22×C10), (C22×C4).124(C2×C10), SmallGroup(320,1532)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C41D4
C1C2C22C2×C10C22×C10D4×C10C5×C41D4 — C10×C41D4
C1C22 — C10×C41D4
C1C22×C10 — C10×C41D4

Generators and relations for C10×C41D4
 G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 882 in 498 conjugacy classes, 210 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, C23, C10, C10, C42, C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2×C42, C41D4, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C41D4, C4×C20, C22×C20, D4×C10, D4×C10, C23×C10, C2×C4×C20, C5×C41D4, D4×C2×C10, C10×C41D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C41D4, C22×D4, C5×D4, C22×C10, C2×C41D4, D4×C10, C23×C10, C5×C41D4, D4×C2×C10, C10×C41D4

Smallest permutation representation of C10×C41D4
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 16 62 154)(2 17 63 155)(3 18 64 156)(4 19 65 157)(5 20 66 158)(6 11 67 159)(7 12 68 160)(8 13 69 151)(9 14 70 152)(10 15 61 153)(21 41 40 55)(22 42 31 56)(23 43 32 57)(24 44 33 58)(25 45 34 59)(26 46 35 60)(27 47 36 51)(28 48 37 52)(29 49 38 53)(30 50 39 54)(71 139 99 111)(72 140 100 112)(73 131 91 113)(74 132 92 114)(75 133 93 115)(76 134 94 116)(77 135 95 117)(78 136 96 118)(79 137 97 119)(80 138 98 120)(81 145 105 121)(82 146 106 122)(83 147 107 123)(84 148 108 124)(85 149 109 125)(86 150 110 126)(87 141 101 127)(88 142 102 128)(89 143 103 129)(90 144 104 130)
(1 97 60 107)(2 98 51 108)(3 99 52 109)(4 100 53 110)(5 91 54 101)(6 92 55 102)(7 93 56 103)(8 94 57 104)(9 95 58 105)(10 96 59 106)(11 114 21 128)(12 115 22 129)(13 116 23 130)(14 117 24 121)(15 118 25 122)(16 119 26 123)(17 120 27 124)(18 111 28 125)(19 112 29 126)(20 113 30 127)(31 143 160 133)(32 144 151 134)(33 145 152 135)(34 146 153 136)(35 147 154 137)(36 148 155 138)(37 149 156 139)(38 150 157 140)(39 141 158 131)(40 142 159 132)(41 88 67 74)(42 89 68 75)(43 90 69 76)(44 81 70 77)(45 82 61 78)(46 83 62 79)(47 84 63 80)(48 85 64 71)(49 86 65 72)(50 87 66 73)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 111)(9 112)(10 113)(11 97)(12 98)(13 99)(14 100)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 107)(22 108)(23 109)(24 110)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 81)(39 82)(40 83)(41 147)(42 148)(43 149)(44 150)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 129)(52 130)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

140 conjugacy classes

class 1 2A···2G2H···2O4A···4L5A5B5C5D10A···10AB10AC···10BH20A···20AV
order12···22···24···4555510···1010···1020···20
size11···14···42···211111···14···42···2

140 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C5C10C10C10D4C5×D4
kernelC10×C41D4C2×C4×C20C5×C41D4D4×C2×C10C2×C41D4C2×C42C41D4C22×D4C2×C20C2×C4
# reps11864432241248

Matrix representation of C10×C41D4 in GL5(𝔽41)

400000
01000
00100
000370
000037
,
400000
040200
040100
000342
000167
,
10000
01000
00100
000342
000167
,
400000
013900
004000
000400
000341

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,40,40,0,0,0,2,1,0,0,0,0,0,34,16,0,0,0,2,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,16,0,0,0,2,7],[40,0,0,0,0,0,1,0,0,0,0,39,40,0,0,0,0,0,40,34,0,0,0,0,1] >;

C10×C41D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4\rtimes_1D_4
% in TeX

G:=Group("C10xC4:1D4");
// GroupNames label

G:=SmallGroup(320,1532);
// by ID

G=gap.SmallGroup(320,1532);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,856]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

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