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G = D104Q16order 320 = 26·5

1st semidirect product of D10 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D104Q16, Q8.6D20, Dic10.12D4, C4.93(D4×D5), C4.6(C2×D20), (C5×Q8).1D4, C2.9(D5×Q16), C4⋊C4.25D10, Q8⋊C44D5, (C2×C8).16D10, (C2×Dic20)⋊6C2, C20.122(C2×D4), C10.17(C2×Q16), C52(C22⋊Q16), C10.24C22≀C2, D101C8.2C2, C10.Q1612C2, (C2×C40).16C22, D102Q8.2C2, (C2×Q8).108D10, (C2×Dic5).39D4, C22.198(D4×D5), (C2×C20).248C23, (C22×D5).115D4, (Q8×C10).31C22, C2.27(C22⋊D20), C2.17(SD16⋊D5), C10.35(C8.C22), (C2×Dic10).75C22, (C2×Q8×D5).4C2, (C2×C5⋊Q16)⋊3C2, (C5×Q8⋊C4)⋊4C2, (C2×C4×D5).25C22, (C2×C10).261(C2×D4), (C5×C4⋊C4).49C22, (C2×C52C8).39C22, (C2×C4).355(C22×D5), SmallGroup(320,435)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D104Q16
C1C5C10C2×C10C2×C20C2×C4×D5C2×Q8×D5 — D104Q16
C5C10C2×C20 — D104Q16
C1C22C2×C4Q8⋊C4

Generators and relations for D104Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 606 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, C4⋊C4, C2×C8, C2×C8, Q16 [×4], C22×C4 [×2], C2×Q8, C2×Q8 [×7], Dic5 [×4], C20 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C22⋊Q8, C2×Q16 [×2], C22×Q8, C52C8, C40, Dic10 [×2], Dic10 [×7], C4×D5 [×6], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22⋊Q16, Dic20 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×C10, C10.Q16, D101C8, C5×Q8⋊C4, D102Q8, C2×Dic20, C2×C5⋊Q16, C2×Q8×D5, D104Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, D20 [×2], C22×D5, C22⋊Q16, C2×D20, D4×D5 [×2], C22⋊D20, SD16⋊D5, D5×Q16, D104Q16

Smallest permutation representation of D104Q16
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 21 14 135 33 143 50 158)(2 30 15 134 34 142 41 157)(3 29 16 133 35 141 42 156)(4 28 17 132 36 150 43 155)(5 27 18 131 37 149 44 154)(6 26 19 140 38 148 45 153)(7 25 20 139 39 147 46 152)(8 24 11 138 40 146 47 151)(9 23 12 137 31 145 48 160)(10 22 13 136 32 144 49 159)(51 124 81 117 74 102 69 94)(52 123 82 116 75 101 70 93)(53 122 83 115 76 110 61 92)(54 121 84 114 77 109 62 91)(55 130 85 113 78 108 63 100)(56 129 86 112 79 107 64 99)(57 128 87 111 80 106 65 98)(58 127 88 120 71 105 66 97)(59 126 89 119 72 104 67 96)(60 125 90 118 73 103 68 95)
(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), (1,21,14,135,33,143,50,158)(2,30,15,134,34,142,41,157)(3,29,16,133,35,141,42,156)(4,28,17,132,36,150,43,155)(5,27,18,131,37,149,44,154)(6,26,19,140,38,148,45,153)(7,25,20,139,39,147,46,152)(8,24,11,138,40,146,47,151)(9,23,12,137,31,145,48,160)(10,22,13,136,32,144,49,159)(51,124,81,117,74,102,69,94)(52,123,82,116,75,101,70,93)(53,122,83,115,76,110,61,92)(54,121,84,114,77,109,62,91)(55,130,85,113,78,108,63,100)(56,129,86,112,79,107,64,99)(57,128,87,111,80,106,65,98)(58,127,88,120,71,105,66,97)(59,126,89,119,72,104,67,96)(60,125,90,118,73,103,68,95), (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), (1,21,14,135,33,143,50,158)(2,30,15,134,34,142,41,157)(3,29,16,133,35,141,42,156)(4,28,17,132,36,150,43,155)(5,27,18,131,37,149,44,154)(6,26,19,140,38,148,45,153)(7,25,20,139,39,147,46,152)(8,24,11,138,40,146,47,151)(9,23,12,137,31,145,48,160)(10,22,13,136,32,144,49,159)(51,124,81,117,74,102,69,94)(52,123,82,116,75,101,70,93)(53,122,83,115,76,110,61,92)(54,121,84,114,77,109,62,91)(55,130,85,113,78,108,63,100)(56,129,86,112,79,107,64,99)(57,128,87,111,80,106,65,98)(58,127,88,120,71,105,66,97)(59,126,89,119,72,104,67,96)(60,125,90,118,73,103,68,95), (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)], [(1,21,14,135,33,143,50,158),(2,30,15,134,34,142,41,157),(3,29,16,133,35,141,42,156),(4,28,17,132,36,150,43,155),(5,27,18,131,37,149,44,154),(6,26,19,140,38,148,45,153),(7,25,20,139,39,147,46,152),(8,24,11,138,40,146,47,151),(9,23,12,137,31,145,48,160),(10,22,13,136,32,144,49,159),(51,124,81,117,74,102,69,94),(52,123,82,116,75,101,70,93),(53,122,83,115,76,110,61,92),(54,121,84,114,77,109,62,91),(55,130,85,113,78,108,63,100),(56,129,86,112,79,107,64,99),(57,128,87,111,80,106,65,98),(58,127,88,120,71,105,66,97),(59,126,89,119,72,104,67,96),(60,125,90,118,73,103,68,95)], [(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)])

47 conjugacy classes

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

47 irreducible representations

dim11111111222222222244444
type+++++++++++++-++++-++--
imageC1C2C2C2C2C2C2C2D4D4D4D4D5Q16D10D10D10D20C8.C22D4×D5D4×D5SD16⋊D5D5×Q16
kernelD104Q16C10.Q16D101C8C5×Q8⋊C4D102Q8C2×Dic20C2×C5⋊Q16C2×Q8×D5Dic10C2×Dic5C5×Q8C22×D5Q8⋊C4D10C4⋊C4C2×C8C2×Q8Q8C10C4C22C2C2
# reps11111111212124222812244

Matrix representation of D104Q16 in GL6(𝔽41)

35350000
6400000
0040000
0004000
000010
000001
,
35350000
4060000
0040000
0027100
0000400
0000040
,
100000
6400000
00283700
00221300
000007
00003524
,
4000000
3510000
00283700
0011300
0000313
00002110

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],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,27,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,28,22,0,0,0,0,37,13,0,0,0,0,0,0,0,35,0,0,0,0,7,24],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,28,1,0,0,0,0,37,13,0,0,0,0,0,0,31,21,0,0,0,0,3,10] >;

D104Q16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_4Q_{16}
% in TeX

G:=Group("D10:4Q16");
// GroupNames label

G:=SmallGroup(320,435);
// by ID

G=gap.SmallGroup(320,435);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,851,438,102,12550]);
// Polycyclic

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

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