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G = SD16×C20order 320 = 26·5

Direct product of C20 and SD16

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

Aliases: SD16×C20, C85(C2×C20), (C4×C40)⋊27C2, C4037(C2×C4), (C4×C8)⋊11C10, (C4×Q8)⋊1C10, Q81(C2×C20), (Q8×C20)⋊21C2, D4.1(C2×C20), (C4×D4).4C10, C4.Q814C10, C2.13(D4×C20), (D4×C20).19C2, (C2×C20).361D4, C10.145(C4×D4), C2.4(C10×SD16), Q8⋊C421C10, D4⋊C4.8C10, C42.71(C2×C10), C4.10(C22×C20), (C2×SD16).5C10, C10.84(C2×SD16), C22.52(D4×C10), C20.257(C4○D4), C10.117(C4○D8), C20.214(C22×C4), (C2×C20).905C23, (C2×C40).437C22, (C4×C20).356C22, (C10×SD16).10C2, (D4×C10).291C22, (Q8×C10).254C22, C2.4(C5×C4○D8), C4.2(C5×C4○D4), (C5×Q8)⋊22(C2×C4), (C5×C4.Q8)⋊29C2, (C2×C4).51(C5×D4), C4⋊C4.46(C2×C10), (C2×C8).66(C2×C10), (C5×D4).32(C2×C4), (C5×Q8⋊C4)⋊44C2, (C2×D4).49(C2×C10), (C2×C10).628(C2×D4), (C2×Q8).39(C2×C10), (C5×D4⋊C4).17C2, (C5×C4⋊C4).367C22, (C2×C4).80(C22×C10), SmallGroup(320,939)

Series: Derived Chief Lower central Upper central

C1C4 — SD16×C20
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×Q8⋊C4 — SD16×C20
C1C2C4 — SD16×C20
C1C2×C20C4×C20 — SD16×C20

Generators and relations for SD16×C20
 G = < a,b,c | a20=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 202 in 122 conjugacy classes, 74 normal (50 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], 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 [×2], C20 [×5], C2×C10, C2×C10 [×4], C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, 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, C4×SD16, 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, C4×C40, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.Q8, D4×C20, Q8×C20, C10×SD16, SD16×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], SD16 [×2], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×SD16, C4○D8, C2×C20 [×6], C5×D4 [×2], C22×C10, C4×SD16, C5×SD16 [×2], C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×SD16, C5×C4○D8, SD16×C20

Smallest permutation representation of SD16×C20
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)
(1 133 72 37 110 90 144 57)(2 134 73 38 111 91 145 58)(3 135 74 39 112 92 146 59)(4 136 75 40 113 93 147 60)(5 137 76 21 114 94 148 41)(6 138 77 22 115 95 149 42)(7 139 78 23 116 96 150 43)(8 140 79 24 117 97 151 44)(9 121 80 25 118 98 152 45)(10 122 61 26 119 99 153 46)(11 123 62 27 120 100 154 47)(12 124 63 28 101 81 155 48)(13 125 64 29 102 82 156 49)(14 126 65 30 103 83 157 50)(15 127 66 31 104 84 158 51)(16 128 67 32 105 85 159 52)(17 129 68 33 106 86 160 53)(18 130 69 34 107 87 141 54)(19 131 70 35 108 88 142 55)(20 132 71 36 109 89 143 56)
(21 137)(22 138)(23 139)(24 140)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B5C5D8A···8H10A···10L10M···10T20A···20P20Q···20AF20AG···20BD40A···40AF
order122222444444444···455558···810···1010···1020···2020···2020···2040···40
size111144111122224···411112···21···14···41···12···24···42···2

140 irreducible representations

dim11111111111111111122222222
type+++++++++
imageC1C2C2C2C2C2C2C2C4C5C10C10C10C10C10C10C10C20D4SD16C4○D4C4○D8C5×D4C5×SD16C5×C4○D4C5×C4○D8
kernelSD16×C20C4×C40C5×D4⋊C4C5×Q8⋊C4C5×C4.Q8D4×C20Q8×C20C10×SD16C5×SD16C4×SD16C4×C8D4⋊C4Q8⋊C4C4.Q8C4×D4C4×Q8C2×SD16SD16C2×C20C20C20C10C2×C4C4C4C2
# reps11111111844444444322424816816

Matrix representation of SD16×C20 in GL3(𝔽41) generated by

3200
0180
0018
,
4000
01515
02615
,
4000
010
0040
G:=sub<GL(3,GF(41))| [32,0,0,0,18,0,0,0,18],[40,0,0,0,15,26,0,15,15],[40,0,0,0,1,0,0,0,40] >;

SD16×C20 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{20}
% in TeX

G:=Group("SD16xC20");
// GroupNames label

G:=SmallGroup(320,939);
// by ID

G=gap.SmallGroup(320,939);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,436,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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