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G = D102Q16order 320 = 26·5

2nd semidirect product of D10 and Q16 acting via Q16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102Q16, C40.17D4, C8.19D20, C2.D86D5, C4⋊C4.52D10, C4.55(C2×D20), C2.15(D5×Q16), C20.135(C2×D4), (C2×C8).232D10, C52(C8.18D4), C10.25(C2×Q16), (C2×Dic20)⋊16C2, C20.42(C4○D4), C10.29(C4○D8), C10.Q1621C2, (C2×C40).84C22, D102Q8.9C2, (C22×D5).87D4, C22.233(D4×D5), C2.14(D83D5), C2.21(C4⋊D20), C10.48(C4⋊D4), (C2×C20).303C23, C4.11(Q82D5), (C2×Dic5).149D4, (C2×Dic10).95C22, (D5×C2×C8).4C2, (C5×C2.D8)⋊6C2, (C2×C10).308(C2×D4), (C5×C4⋊C4).96C22, (C2×C4×D5).307C22, (C2×C4).406(C22×D5), (C2×C52C8).244C22, SmallGroup(320,514)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D102Q16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — D102Q16
C5C10C2×C20 — D102Q16
C1C22C2×C4C2.D8

Generators and relations for D102Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 454 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], Q8 [×4], C23, D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×3], Q16 [×2], C22×C4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, Q8⋊C4 [×2], C2.D8, C22⋊Q8 [×2], C22×C8, C2×Q16, C52C8, C40 [×2], Dic10 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C8.18D4, C8×D5 [×2], Dic20 [×2], C2×C52C8, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10 [×2], C2×C4×D5, C10.Q16 [×2], C5×C2.D8, D102Q8 [×2], D5×C2×C8, C2×Dic20, D102Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, Q16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×Q16, C4○D8, D20 [×2], C22×D5, C8.18D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D83D5, D5×Q16, D102Q16

Smallest permutation representation of D102Q16
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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 150)(29 149)(30 148)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)(61 81)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(91 111)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 130)(108 129)(109 128)(110 127)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 160)(138 159)(139 158)(140 157)
(1 130 50 113 33 108 14 100)(2 121 41 114 34 109 15 91)(3 122 42 115 35 110 16 92)(4 123 43 116 36 101 17 93)(5 124 44 117 37 102 18 94)(6 125 45 118 38 103 19 95)(7 126 46 119 39 104 20 96)(8 127 47 120 40 105 11 97)(9 128 48 111 31 106 12 98)(10 129 49 112 32 107 13 99)(21 85 135 55 143 63 158 78)(22 86 136 56 144 64 159 79)(23 87 137 57 145 65 160 80)(24 88 138 58 146 66 151 71)(25 89 139 59 147 67 152 72)(26 90 140 60 148 68 153 73)(27 81 131 51 149 69 154 74)(28 82 132 52 150 70 155 75)(29 83 133 53 141 61 156 76)(30 84 134 54 142 62 157 77)
(1 55 33 78)(2 54 34 77)(3 53 35 76)(4 52 36 75)(5 51 37 74)(6 60 38 73)(7 59 39 72)(8 58 40 71)(9 57 31 80)(10 56 32 79)(11 66 47 88)(12 65 48 87)(13 64 49 86)(14 63 50 85)(15 62 41 84)(16 61 42 83)(17 70 43 82)(18 69 44 81)(19 68 45 90)(20 67 46 89)(21 100 143 113)(22 99 144 112)(23 98 145 111)(24 97 146 120)(25 96 147 119)(26 95 148 118)(27 94 149 117)(28 93 150 116)(29 92 141 115)(30 91 142 114)(101 155 123 132)(102 154 124 131)(103 153 125 140)(104 152 126 139)(105 151 127 138)(106 160 128 137)(107 159 129 136)(108 158 130 135)(109 157 121 134)(110 156 122 133)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444558888888810···102020202020···2040···40
size11111010228810104040222222101010102···244448···84···4

50 irreducible representations

dim11111122222222224444
type++++++++++-+++++--
imageC1C2C2C2C2C2D4D4D4D5C4○D4Q16D10D10C4○D8D20Q82D5D4×D5D83D5D5×Q16
kernelD102Q16C10.Q16C5×C2.D8D102Q8D5×C2×C8C2×Dic20C40C2×Dic5C22×D5C2.D8C20D10C4⋊C4C2×C8C10C8C4C22C2C2
# reps12121121122442482244

Matrix representation of D102Q16 in GL4(𝔽41) generated by

1600
35600
00400
00040
,
40000
6100
0010
003040
,
1000
0100
00140
0013
,
252500
391600
001937
002922
G:=sub<GL(4,GF(41))| [1,35,0,0,6,6,0,0,0,0,40,0,0,0,0,40],[40,6,0,0,0,1,0,0,0,0,1,30,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,14,1,0,0,0,3],[25,39,0,0,25,16,0,0,0,0,19,29,0,0,37,22] >;

D102Q16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_2Q_{16}
% in TeX

G:=Group("D10:2Q16");
// GroupNames label

G:=SmallGroup(320,514);
// by ID

G=gap.SmallGroup(320,514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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