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G = C5×C4○D16order 320 = 26·5

Direct product of C5 and C4○D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×C4○D16, D163C10, Q323C10, C40.76D4, C20.71D8, SD323C10, C80.23C22, C40.75C23, (C2×C80)⋊13C2, (C2×C16)⋊6C10, C4○D81C10, (C5×D16)⋊7C2, (C5×Q32)⋊7C2, C4.20(C5×D8), C8.13(C5×D4), C16.6(C2×C10), (C5×SD32)⋊7C2, D8.2(C2×C10), C10.87(C2×D8), C4.10(D4×C10), C2.15(C10×D8), (C2×C10).12D8, C22.1(C5×D8), C20.317(C2×D4), (C2×C20).429D4, C8.6(C22×C10), Q16.2(C2×C10), (C5×D8).12C22, (C2×C40).434C22, (C5×Q16).14C22, (C5×C4○D8)⋊8C2, (C2×C4).85(C5×D4), (C2×C8).91(C2×C10), SmallGroup(320,1009)

Series: Derived Chief Lower central Upper central

C1C8 — C5×C4○D16
C1C2C4C8C40C5×D8C5×D16 — C5×C4○D16
C1C2C4C8 — C5×C4○D16
C1C20C2×C20C2×C40 — C5×C4○D16

Generators and relations for C5×C4○D16
 G = < a,b,c,d | a5=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C10, C10 [×3], C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×4], C5×Q8 [×2], C4○D16, C80 [×2], C2×C40, C5×D8 [×2], C5×SD16 [×2], C5×Q16 [×2], C5×C4○D4 [×2], C2×C80, C5×D16, C5×SD32 [×2], C5×Q32, C5×C4○D8 [×2], C5×C4○D16
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], C2×D8, C5×D4 [×2], C22×C10, C4○D16, C5×D8 [×2], D4×C10, C10×D8, C5×C4○D16

Smallest permutation representation of C5×C4○D16
On 160 points
Generators in S160
(1 128 38 70 53)(2 113 39 71 54)(3 114 40 72 55)(4 115 41 73 56)(5 116 42 74 57)(6 117 43 75 58)(7 118 44 76 59)(8 119 45 77 60)(9 120 46 78 61)(10 121 47 79 62)(11 122 48 80 63)(12 123 33 65 64)(13 124 34 66 49)(14 125 35 67 50)(15 126 36 68 51)(16 127 37 69 52)(17 92 153 99 138)(18 93 154 100 139)(19 94 155 101 140)(20 95 156 102 141)(21 96 157 103 142)(22 81 158 104 143)(23 82 159 105 144)(24 83 160 106 129)(25 84 145 107 130)(26 85 146 108 131)(27 86 147 109 132)(28 87 148 110 133)(29 88 149 111 134)(30 89 150 112 135)(31 90 151 97 136)(32 91 152 98 137)
(1 107 9 99)(2 108 10 100)(3 109 11 101)(4 110 12 102)(5 111 13 103)(6 112 14 104)(7 97 15 105)(8 98 16 106)(17 38 25 46)(18 39 26 47)(19 40 27 48)(20 41 28 33)(21 42 29 34)(22 43 30 35)(23 44 31 36)(24 45 32 37)(49 157 57 149)(50 158 58 150)(51 159 59 151)(52 160 60 152)(53 145 61 153)(54 146 62 154)(55 147 63 155)(56 148 64 156)(65 95 73 87)(66 96 74 88)(67 81 75 89)(68 82 76 90)(69 83 77 91)(70 84 78 92)(71 85 79 93)(72 86 80 94)(113 131 121 139)(114 132 122 140)(115 133 123 141)(116 134 124 142)(117 135 125 143)(118 136 126 144)(119 137 127 129)(120 138 128 130)
(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)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)(49 56)(50 55)(51 54)(52 53)(57 64)(58 63)(59 62)(60 61)(65 74)(66 73)(67 72)(68 71)(69 70)(75 80)(76 79)(77 78)(81 86)(82 85)(83 84)(87 96)(88 95)(89 94)(90 93)(91 92)(97 100)(98 99)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 128)(129 130)(131 144)(132 143)(133 142)(134 141)(135 140)(136 139)(137 138)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H10I···10P16A···16H20A···20H20I20J20K20L20M···20T40A···40P80A···80AF
order122224444455558888101010101010101010···1016···1620···202020202020···2040···4080···80
size112881128811112222111122228···82···21···122228···82···22···2

110 irreducible representations

dim1111111111112222222222
type++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4D8D8C5×D4C5×D4C4○D16C5×D8C5×D8C5×C4○D16
kernelC5×C4○D16C2×C80C5×D16C5×SD32C5×Q32C5×C4○D8C4○D16C2×C16D16SD32Q32C4○D8C40C2×C20C20C2×C10C8C2×C4C5C4C22C1
# reps11121244484811224488832

Matrix representation of C5×C4○D16 in GL2(𝔽241) generated by

910
091
,
1770
0177
,
183187
27129
,
183187
15658
G:=sub<GL(2,GF(241))| [91,0,0,91],[177,0,0,177],[183,27,187,129],[183,156,187,58] >;

C5×C4○D16 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_{16}
% in TeX

G:=Group("C5xC4oD16");
// GroupNames label

G:=SmallGroup(320,1009);
// by ID

G=gap.SmallGroup(320,1009);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,856,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations

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