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G = D103Q16order 320 = 26·5

3rd semidirect product of D10 and Q16 acting via Q16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D103Q16, C40.27D4, (C2×Q16)⋊5D5, (C10×Q16)⋊5C2, C405C424C2, C2.19(D5×Q16), (C2×C8).243D10, C20.186(C2×D4), C8.28(C5⋊D4), C55(C8.18D4), (C2×Q8).64D10, C10.30(C2×Q16), C10.81(C4○D8), Q8⋊Dic535C2, (C2×C40).95C22, D103Q8.9C2, (C22×D5).91D4, C22.279(D4×D5), C20.107(C4○D4), C4.36(D42D5), C2.22(C202D4), (C2×C20).462C23, (C2×Dic5).162D4, (Q8×C10).91C22, C2.18(Q8.D10), C10.121(C4⋊D4), C4⋊Dic5.185C22, (D5×C2×C8).5C2, C4.85(C2×C5⋊D4), (C2×C10).373(C2×D4), (C2×C4×D5).311C22, (C2×C4).550(C22×D5), (C2×C52C8).286C22, SmallGroup(320,815)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D103Q16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — D103Q16
C5C10C2×C20 — D103Q16
C1C22C2×C4C2×Q16

Generators and relations for D103Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 406 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 [×4], 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], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C8.18D4, C8×D5 [×2], C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40, C5×Q16 [×2], C2×C4×D5, Q8×C10 [×2], C405C4, Q8⋊Dic5 [×2], D5×C2×C8, D103Q8 [×2], C10×Q16, D103Q16
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, C5⋊D4 [×2], C22×D5, C8.18D4, D4×D5, D42D5, C2×C5⋊D4, D5×Q16, Q8.D10, C202D4, D103Q16

Smallest permutation representation of D103Q16
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)(21 149)(22 148)(23 147)(24 146)(25 145)(26 144)(27 143)(28 142)(29 141)(30 150)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 83)(62 82)(63 81)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(91 115)(92 114)(93 113)(94 112)(95 111)(96 120)(97 119)(98 118)(99 117)(100 116)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 130)(110 129)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)
(1 130 50 113 33 110 14 94)(2 121 41 114 34 101 15 95)(3 122 42 115 35 102 16 96)(4 123 43 116 36 103 17 97)(5 124 44 117 37 104 18 98)(6 125 45 118 38 105 19 99)(7 126 46 119 39 106 20 100)(8 127 47 120 40 107 11 91)(9 128 48 111 31 108 12 92)(10 129 49 112 32 109 13 93)(21 85 139 59 145 65 158 78)(22 86 140 60 146 66 159 79)(23 87 131 51 147 67 160 80)(24 88 132 52 148 68 151 71)(25 89 133 53 149 69 152 72)(26 90 134 54 150 70 153 73)(27 81 135 55 141 61 154 74)(28 82 136 56 142 62 155 75)(29 83 137 57 143 63 156 76)(30 84 138 58 144 64 157 77)
(1 73 33 54)(2 74 34 55)(3 75 35 56)(4 76 36 57)(5 77 37 58)(6 78 38 59)(7 79 39 60)(8 80 40 51)(9 71 31 52)(10 72 32 53)(11 87 47 67)(12 88 48 68)(13 89 49 69)(14 90 50 70)(15 81 41 61)(16 82 42 62)(17 83 43 63)(18 84 44 64)(19 85 45 65)(20 86 46 66)(21 118 145 99)(22 119 146 100)(23 120 147 91)(24 111 148 92)(25 112 149 93)(26 113 150 94)(27 114 141 95)(28 115 142 96)(29 116 143 97)(30 117 144 98)(101 135 121 154)(102 136 122 155)(103 137 123 156)(104 138 124 157)(105 139 125 158)(106 140 126 159)(107 131 127 160)(108 132 128 151)(109 133 129 152)(110 134 130 153)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim11111122222222224444
type++++++++++-++-+-+
imageC1C2C2C2C2C2D4D4D4D5C4○D4Q16D10D10C4○D8C5⋊D4D42D5D4×D5D5×Q16Q8.D10
kernelD103Q16C405C4Q8⋊Dic5D5×C2×C8D103Q8C10×Q16C40C2×Dic5C22×D5C2×Q16C20D10C2×C8C2×Q8C10C8C4C22C2C2
# reps11212121122424482244

Matrix representation of D103Q16 in GL4(𝔽41) generated by

6600
35100
00400
00040
,
6600
13500
0010
002640
,
1000
0100
00270
002138
,
183500
62300
001218
001729
G:=sub<GL(4,GF(41))| [6,35,0,0,6,1,0,0,0,0,40,0,0,0,0,40],[6,1,0,0,6,35,0,0,0,0,1,26,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,27,21,0,0,0,38],[18,6,0,0,35,23,0,0,0,0,12,17,0,0,18,29] >;

D103Q16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_3Q_{16}
% in TeX

G:=Group("D10:3Q16");
// GroupNames label

G:=SmallGroup(320,815);
// by ID

G=gap.SmallGroup(320,815);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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