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G = D10⋊C16order 320 = 26·5

2nd semidirect product of D10 and C16 acting via C16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102C16, C10.2M5(2), C20.13M4(2), (C2×C8).7F5, (C2×C40).4C4, C51(C22⋊C16), C10.4(C2×C16), C52C8.53D4, C2.4(D5⋊C16), (C22×D5).6C8, C4.11(C4.F5), C10.3(C22⋊C8), C22.9(D5⋊C8), (C2×Dic5).10C8, C4.35(C22⋊F5), C2.2(C8.F5), C2.1(D10⋊C8), C20.33(C22⋊C4), (C2×C5⋊C16)⋊5C2, (C2×C4×D5).41C4, (D5×C2×C8).10C2, (C2×C10).5(C2×C8), (C2×C4).155(C2×F5), (C2×C20).161(C2×C4), (C2×C52C8).345C22, SmallGroup(320,225)

Series: Derived Chief Lower central Upper central

C1C10 — D10⋊C16
C1C5C10C20C52C8C2×C52C8C2×C5⋊C16 — D10⋊C16
C5C10 — D10⋊C16
C1C2×C4C2×C8

Generators and relations for D10⋊C16
 G = < a,b,c | a10=b2=c16=1, bab=a-1, cac-1=a3, cbc-1=a7b >

Subgroups: 226 in 66 conjugacy classes, 30 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4, C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×3], C23, D5 [×2], C10 [×3], C16 [×2], C2×C8, C2×C8 [×3], C22×C4, Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], C22×C8, C52C8 [×2], C40, C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C22⋊C16, C5⋊C16 [×2], C8×D5 [×2], C2×C52C8, C2×C40, C2×C4×D5, C2×C5⋊C16 [×2], D5×C2×C8, D10⋊C16
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C16 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×C16, M5(2), C2×F5, C22⋊C16, D5⋊C8, C4.F5, C22⋊F5, D5⋊C16, C8.F5, D10⋊C8, D10⋊C16

Smallest permutation representation of D10⋊C16
On 160 points
Generators in S160
(1 72 40 138 124 94 62 104 151 29)(2 139 63 30 41 95 152 73 125 105)(3 31 153 106 64 96 126 140 42 74)(4 107 127 75 154 81 43 32 49 141)(5 76 44 142 128 82 50 108 155 17)(6 143 51 18 45 83 156 77 113 109)(7 19 157 110 52 84 114 144 46 78)(8 111 115 79 158 85 47 20 53 129)(9 80 48 130 116 86 54 112 159 21)(10 131 55 22 33 87 160 65 117 97)(11 23 145 98 56 88 118 132 34 66)(12 99 119 67 146 89 35 24 57 133)(13 68 36 134 120 90 58 100 147 25)(14 135 59 26 37 91 148 69 121 101)(15 27 149 102 60 92 122 136 38 70)(16 103 123 71 150 93 39 28 61 137)
(1 21)(2 33)(3 66)(4 146)(5 25)(6 37)(7 70)(8 150)(9 29)(10 41)(11 74)(12 154)(13 17)(14 45)(15 78)(16 158)(18 135)(19 38)(20 28)(22 139)(23 42)(24 32)(26 143)(27 46)(30 131)(31 34)(35 49)(36 108)(39 53)(40 112)(43 57)(44 100)(47 61)(48 104)(50 134)(51 59)(52 92)(54 138)(55 63)(56 96)(58 142)(60 84)(62 130)(64 88)(65 73)(67 107)(68 155)(69 77)(71 111)(72 159)(75 99)(76 147)(79 103)(80 151)(81 133)(82 120)(83 101)(85 137)(86 124)(87 105)(89 141)(90 128)(91 109)(93 129)(94 116)(95 97)(98 126)(102 114)(106 118)(110 122)(113 148)(115 123)(117 152)(119 127)(121 156)(125 160)(132 153)(136 157)(140 145)(144 149)
(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)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F 5 8A8B8C8D8E···8L10A10B10C16A···16P20A20B20C20D40A···40H
order122222444444588888···810101016···162020202040···40
size1111101011111010422225···544410···1044444···4

56 irreducible representations

dim111111112224444444
type+++++++
imageC1C2C2C4C4C8C8C16D4M4(2)M5(2)F5C2×F5C4.F5C22⋊F5D5⋊C8D5⋊C16C8.F5
kernelD10⋊C16C2×C5⋊C16D5×C2×C8C2×C40C2×C4×D5C2×Dic5C22×D5D10C52C8C20C10C2×C8C2×C4C4C4C22C2C2
# reps1212244162241122244

Matrix representation of D10⋊C16 in GL8(𝔽241)

2400000000
0240000000
0024000000
0002400000
0000002401
0000002400
0000102400
0000012400
,
10000000
165240000000
0024000000
0023910000
0000012400
0000102400
0000002400
0000002401
,
165239000000
076000000
0012400000
0022400000
00001141271410
0000141270114
0000114012714
00000141127114

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0],[1,165,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,239,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1],[165,0,0,0,0,0,0,0,239,76,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,114,14,114,0,0,0,0,0,127,127,0,141,0,0,0,0,141,0,127,127,0,0,0,0,0,114,14,114] >;

D10⋊C16 in GAP, Magma, Sage, TeX

D_{10}\rtimes C_{16}
% in TeX

G:=Group("D10:C16");
// GroupNames label

G:=SmallGroup(320,225);
// by ID

G=gap.SmallGroup(320,225);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,100,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^10=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^7*b>;
// generators/relations

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