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G = C5×SD16⋊C4order 320 = 26·5

Direct product of C5 and SD16⋊C4

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

Aliases: C5×SD16⋊C4, SD161C20, C82(C2×C20), C4030(C2×C4), (C4×Q8)⋊2C10, Q82(C2×C20), C8⋊C41C10, (Q8×C20)⋊22C2, (C5×SD16)⋊9C4, D4.2(C2×C20), (C4×D4).5C10, C2.D811C10, C2.15(D4×C20), (D4×C20).20C2, C10.147(C4×D4), (C2×C20).456D4, C42.8(C2×C10), Q8⋊C416C10, D4⋊C4.6C10, C4.12(C22×C20), (C10×SD16).4C2, (C2×SD16).1C10, C22.54(D4×C10), C20.259(C4○D4), C20.216(C22×C4), (C2×C40).329C22, (C2×C20).907C23, (C4×C20).249C22, C10.129(C8⋊C22), (D4×C10).292C22, (Q8×C10).256C22, C10.129(C8.C22), C4.4(C5×C4○D4), (C5×Q8)⋊23(C2×C4), (C5×C2.D8)⋊26C2, (C5×C8⋊C4)⋊10C2, C2.4(C5×C8⋊C22), C4⋊C4.48(C2×C10), (C2×C8).18(C2×C10), (C5×D4).33(C2×C4), (C2×C4).102(C5×D4), C2.4(C5×C8.C22), (C5×Q8⋊C4)⋊39C2, (C2×D4).50(C2×C10), (C2×C10).630(C2×D4), (C2×Q8).41(C2×C10), (C5×D4⋊C4).15C2, (C5×C4⋊C4).369C22, (C2×C4).82(C22×C10), SmallGroup(320,941)

Series: Derived Chief Lower central Upper central

C1C4 — C5×SD16⋊C4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×D4⋊C4 — C5×SD16⋊C4
C1C2C4 — C5×SD16⋊C4
C1C2×C10C4×C20 — C5×SD16⋊C4

Generators and relations for C5×SD16⋊C4
 G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd-1=b5, cd=dc >

Subgroups: 202 in 120 conjugacy classes, 70 normal (50 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C10 [×3], C10 [×2], C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C20 [×2], C20 [×6], C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C40 [×2], C40, C2×C20 [×3], C2×C20 [×5], C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×C10, SD16⋊C4, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C5×SD16 [×4], C22×C20, D4×C10, Q8×C10, C5×C8⋊C4, C5×D4⋊C4, C5×Q8⋊C4, C5×C2.D8, D4×C20, Q8×C20, C10×SD16, C5×SD16⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C8⋊C22, C8.C22, C2×C20 [×6], C5×D4 [×2], C22×C10, SD16⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8⋊C22, C5×C8.C22, C5×SD16⋊C4

Smallest permutation representation of C5×SD16⋊C4
On 160 points
Generators in S160
(1 122 155 40 147)(2 123 156 33 148)(3 124 157 34 149)(4 125 158 35 150)(5 126 159 36 151)(6 127 160 37 152)(7 128 153 38 145)(8 121 154 39 146)(9 116 140 17 132)(10 117 141 18 133)(11 118 142 19 134)(12 119 143 20 135)(13 120 144 21 136)(14 113 137 22 129)(15 114 138 23 130)(16 115 139 24 131)(25 92 49 88 43)(26 93 50 81 44)(27 94 51 82 45)(28 95 52 83 46)(29 96 53 84 47)(30 89 54 85 48)(31 90 55 86 41)(32 91 56 87 42)(57 74 107 66 99)(58 75 108 67 100)(59 76 109 68 101)(60 77 110 69 102)(61 78 111 70 103)(62 79 112 71 104)(63 80 105 72 97)(64 73 106 65 98)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 35)(34 38)(37 39)(41 43)(42 46)(45 47)(49 55)(51 53)(52 56)(57 61)(58 64)(60 62)(65 67)(66 70)(69 71)(73 75)(74 78)(77 79)(82 84)(83 87)(86 88)(90 92)(91 95)(94 96)(98 100)(99 103)(102 104)(106 108)(107 111)(110 112)(113 117)(114 120)(116 118)(121 127)(123 125)(124 128)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)(145 149)(146 152)(148 150)(153 157)(154 160)(156 158)
(1 115 44 101)(2 120 45 98)(3 117 46 103)(4 114 47 100)(5 119 48 97)(6 116 41 102)(7 113 42 99)(8 118 43 104)(9 86 69 152)(10 83 70 149)(11 88 71 146)(12 85 72 151)(13 82 65 148)(14 87 66 145)(15 84 67 150)(16 81 68 147)(17 90 77 160)(18 95 78 157)(19 92 79 154)(20 89 80 159)(21 94 73 156)(22 91 74 153)(23 96 75 158)(24 93 76 155)(25 62 121 142)(26 59 122 139)(27 64 123 144)(28 61 124 141)(29 58 125 138)(30 63 126 143)(31 60 127 140)(32 57 128 137)(33 136 51 106)(34 133 52 111)(35 130 53 108)(36 135 54 105)(37 132 55 110)(38 129 56 107)(39 134 49 112)(40 131 50 109)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G···4L5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20X20Y···20AV40A···40P
order1222224···44···45555888810···1010···1020···2020···2040···40
size1111442···24···4111144441···14···42···24···44···4

110 irreducible representations

dim11111111111111111122224444
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C5C10C10C10C10C10C10C10C20D4C4○D4C5×D4C5×C4○D4C8⋊C22C8.C22C5×C8⋊C22C5×C8.C22
kernelC5×SD16⋊C4C5×C8⋊C4C5×D4⋊C4C5×Q8⋊C4C5×C2.D8D4×C20Q8×C20C10×SD16C5×SD16SD16⋊C4C8⋊C4D4⋊C4Q8⋊C4C2.D8C4×D4C4×Q8C2×SD16SD16C2×C20C20C2×C4C4C10C10C2C2
# reps111111118444444443222881144

Matrix representation of C5×SD16⋊C4 in GL6(𝔽41)

100000
010000
0037000
0003700
0000370
0000037
,
2230000
39390000
005361328
00551313
002813365
0028283636
,
100000
23400000
001000
0004000
000010
0000040
,
900000
090000
000010
000001
001000
000100

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[2,39,0,0,0,0,23,39,0,0,0,0,0,0,5,5,28,28,0,0,36,5,13,28,0,0,13,13,36,36,0,0,28,13,5,36],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×SD16⋊C4 in GAP, Magma, Sage, TeX

C_5\times {\rm SD}_{16}\rtimes C_4
% in TeX

G:=Group("C5xSD16:C4");
// GroupNames label

G:=SmallGroup(320,941);
// by ID

G=gap.SmallGroup(320,941);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,436,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d^-1=b^5,c*d=d*c>;
// generators/relations

׿
×
𝔽