Copied to
clipboard

G = D105Q16order 320 = 26·5

2nd semidirect product of D10 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D105Q16, Dic10.17D4, (C2×Q16)⋊3D5, C4.67(D4×D5), (C5×Q8).9D4, C20.52(C2×D4), (C2×C8).38D10, C2.18(D5×Q16), (C10×Q16)⋊13C2, (C2×Q8).62D10, C10.29(C2×Q16), C54(C22⋊Q16), Q8.8(C5⋊D4), C10.61C22≀C2, Q8⋊Dic533C2, D103Q8.8C2, (C2×Dic5).84D4, C22.277(D4×D5), D101C8.12C2, C20.44D430C2, (C2×C40).252C22, (C2×C20).460C23, (C22×D5).130D4, (Q8×C10).89C22, C2.29(C23⋊D10), C2.28(Q16⋊D5), C10.78(C8.C22), C4⋊Dic5.183C22, (C2×Dic10).136C22, (C2×Q8×D5).5C2, C4.48(C2×C5⋊D4), (C2×C5⋊Q16)⋊20C2, (C2×C4×D5).56C22, (C2×C10).371(C2×D4), (C2×C4).548(C22×D5), (C2×C52C8).165C22, SmallGroup(320,813)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D105Q16
C1C5C10C2×C10C2×C20C2×C4×D5C2×Q8×D5 — D105Q16
C5C10C2×C20 — D105Q16
C1C22C2×C4C2×Q16

Generators and relations for D105Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 574 in 148 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×11], Q8 [×2], Q8 [×10], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, Q16 [×4], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×6], Dic5 [×4], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16, C2×Q16, C22×Q8, C52C8, C40, Dic10 [×2], Dic10 [×5], C4×D5 [×6], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8 [×3], C22×D5, C22⋊Q16, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×C10 [×2], C20.44D4, D101C8, Q8⋊Dic5, C2×C5⋊Q16, D103Q8, C10×Q16, C2×Q8×D5, D105Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, C5⋊D4 [×2], C22×D5, C22⋊Q16, D4×D5 [×2], C2×C5⋊D4, D5×Q16, Q16⋊D5, C23⋊D10, D105Q16

Smallest permutation representation of D105Q16
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 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 149)(22 148)(23 147)(24 146)(25 145)(26 144)(27 143)(28 142)(29 141)(30 150)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 83)(62 82)(63 81)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 130)(110 129)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 160)(140 159)
(1 150 12 153 33 26 50 132)(2 141 13 154 34 27 41 133)(3 142 14 155 35 28 42 134)(4 143 15 156 36 29 43 135)(5 144 16 157 37 30 44 136)(6 145 17 158 38 21 45 137)(7 146 18 159 39 22 46 138)(8 147 19 160 40 23 47 139)(9 148 20 151 31 24 48 140)(10 149 11 152 32 25 49 131)(51 109 89 91 72 129 69 112)(52 110 90 92 73 130 70 113)(53 101 81 93 74 121 61 114)(54 102 82 94 75 122 62 115)(55 103 83 95 76 123 63 116)(56 104 84 96 77 124 64 117)(57 105 85 97 78 125 65 118)(58 106 86 98 79 126 66 119)(59 107 87 99 80 127 67 120)(60 108 88 100 71 128 68 111)
(1 73 33 52)(2 74 34 53)(3 75 35 54)(4 76 36 55)(5 77 37 56)(6 78 38 57)(7 79 39 58)(8 80 40 59)(9 71 31 60)(10 72 32 51)(11 89 49 69)(12 90 50 70)(13 81 41 61)(14 82 42 62)(15 83 43 63)(16 84 44 64)(17 85 45 65)(18 86 46 66)(19 87 47 67)(20 88 48 68)(21 118 145 97)(22 119 146 98)(23 120 147 99)(24 111 148 100)(25 112 149 91)(26 113 150 92)(27 114 141 93)(28 115 142 94)(29 116 143 95)(30 117 144 96)(101 133 121 154)(102 134 122 155)(103 135 123 156)(104 136 124 157)(105 137 125 158)(106 138 126 159)(107 139 127 160)(108 140 128 151)(109 131 129 152)(110 132 130 153)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111110102244820202040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++-++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D5Q16D10D10C5⋊D4C8.C22D4×D5D4×D5D5×Q16Q16⋊D5
kernelD105Q16C20.44D4D101C8Q8⋊Dic5C2×C5⋊Q16D103Q8C10×Q16C2×Q8×D5Dic10C2×Dic5C5×Q8C22×D5C2×Q16D10C2×C8C2×Q8Q8C10C4C22C2C2
# reps1111111121212424812244

Matrix representation of D105Q16 in GL6(𝔽41)

35350000
6400000
0040000
0004000
000010
000001
,
35350000
4060000
0040000
0040100
0000400
0000040
,
4000000
0400000
0016900
00402500
000006
00003417
,
100000
010000
0016900
00402500
00002017
00001521

G:=sub<GL(6,GF(41))| [35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,40,0,0,0,0,9,25,0,0,0,0,0,0,0,34,0,0,0,0,6,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,40,0,0,0,0,9,25,0,0,0,0,0,0,20,15,0,0,0,0,17,21] >;

D105Q16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_5Q_{16}
% in TeX

G:=Group("D10:5Q16");
// GroupNames label

G:=SmallGroup(320,813);
// by ID

G=gap.SmallGroup(320,813);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,851,438,102,12550]);
// 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,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽