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G = Dic58SD16order 320 = 26·5

3rd semidirect product of Dic5 and SD16 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic58SD16, C88(C4×D5), C55(C4×SD16), C4023(C2×C4), C40⋊C29C4, C4.Q813D5, (C8×Dic5)⋊8C2, C10.79(C4×D4), C2.5(D5×SD16), C4⋊C4.158D10, D20.22(C2×C4), (C2×C8).255D10, C22.81(D4×D5), Dic1015(C2×C4), Dic53Q86C2, D208C4.4C2, D206C4.4C2, C10.51(C4○D8), C20.25(C4○D4), C10.Q1614C2, C20.99(C22×C4), C4.1(Q82D5), C10.34(C2×SD16), C2.9(D208C4), (C2×C40).156C22, (C2×C20).268C23, (C2×Dic5).273D4, (C2×D20).76C22, C2.5(SD163D5), (C2×Dic10).82C22, (C4×Dic5).259C22, C4.40(C2×C4×D5), (C5×C4.Q8)⋊6C2, (C2×C40⋊C2).9C2, (C2×C10).273(C2×D4), (C5×C4⋊C4).61C22, (C2×C4).371(C22×D5), (C2×C52C8).231C22, SmallGroup(320,479)

Series: Derived Chief Lower central Upper central

C1C20 — Dic58SD16
C1C5C10C20C2×C20C4×Dic5D208C4 — Dic58SD16
C5C10C20 — Dic58SD16
C1C22C2×C4C4.Q8

Generators and relations for Dic58SD16
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c3 >

Subgroups: 502 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], D4 [×3], Q8 [×3], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×SD16, C40⋊C2 [×4], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D206C4, C10.Q16, C8×Dic5, C5×C4.Q8, Dic53Q8, D208C4, C2×C40⋊C2, Dic58SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×SD16, C4○D8, C4×D5 [×2], C22×D5, C4×SD16, C2×C4×D5, D4×D5, Q82D5, D208C4, D5×SD16, SD163D5, Dic58SD16

Smallest permutation representation of Dic58SD16
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)
(1 103 6 108)(2 102 7 107)(3 101 8 106)(4 110 9 105)(5 109 10 104)(11 53 16 58)(12 52 17 57)(13 51 18 56)(14 60 19 55)(15 59 20 54)(21 92 26 97)(22 91 27 96)(23 100 28 95)(24 99 29 94)(25 98 30 93)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 130 46 125)(42 129 47 124)(43 128 48 123)(44 127 49 122)(45 126 50 121)(61 152 66 157)(62 151 67 156)(63 160 68 155)(64 159 69 154)(65 158 70 153)(71 142 76 147)(72 141 77 146)(73 150 78 145)(74 149 79 144)(75 148 80 143)(81 132 86 137)(82 131 87 136)(83 140 88 135)(84 139 89 134)(85 138 90 133)
(1 148 40 14 28 140 48 160)(2 149 31 15 29 131 49 151)(3 150 32 16 30 132 50 152)(4 141 33 17 21 133 41 153)(5 142 34 18 22 134 42 154)(6 143 35 19 23 135 43 155)(7 144 36 20 24 136 44 156)(8 145 37 11 25 137 45 157)(9 146 38 12 26 138 46 158)(10 147 39 13 27 139 47 159)(51 96 89 124 69 104 71 116)(52 97 90 125 70 105 72 117)(53 98 81 126 61 106 73 118)(54 99 82 127 62 107 74 119)(55 100 83 128 63 108 75 120)(56 91 84 129 64 109 76 111)(57 92 85 130 65 110 77 112)(58 93 86 121 66 101 78 113)(59 94 87 122 67 102 79 114)(60 95 88 123 68 103 80 115)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 140)(20 139)(31 34)(32 33)(35 40)(36 39)(37 38)(41 50)(42 49)(43 48)(44 47)(45 46)(51 82)(52 81)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 72)(62 71)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(91 102)(92 101)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(111 114)(112 113)(115 120)(116 119)(117 118)(121 130)(122 129)(123 128)(124 127)(125 126)(141 152)(142 151)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444444558888888810···102020202020···2040···40
size11112020224444555510102020222222101010102···244448···84···4

56 irreducible representations

dim111111111222222224444
type++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10D10C4○D8C4×D5Q82D5D4×D5D5×SD16SD163D5
kernelDic58SD16D206C4C10.Q16C8×Dic5C5×C4.Q8Dic53Q8D208C4C2×C40⋊C2C40⋊C2C2×Dic5C4.Q8Dic5C20C4⋊C4C2×C8C10C8C4C22C2C2
# reps111111118224242482244

Matrix representation of Dic58SD16 in GL5(𝔽41)

400000
00100
0403400
000400
000040
,
90000
01000
0344000
00090
00009
,
400000
040000
004000
0001515
0002615
,
10000
01000
0344000
00010
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,1,34,0,0,0,0,40,0,0,0,0,0,9,0,0,0,0,0,9],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,15,26,0,0,0,15,15],[1,0,0,0,0,0,1,34,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;

Dic58SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("Dic5:8SD16");
// GroupNames label

G:=SmallGroup(320,479);
// by ID

G=gap.SmallGroup(320,479);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,268,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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