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G = C10.D16order 320 = 26·5

7th non-split extension by C10 of D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D81Dic5, C40.13D4, C10.12D16, C10.5SD32, C20.11SD16, (C5×D8)⋊7C4, (C2×D8).1D5, C54(C2.D16), C40.54(C2×C4), C405C419C2, (C10×D8).2C2, (C2×C10).36D8, C8.7(C2×Dic5), (C2×C20).114D4, (C2×C8).223D10, C8.23(C5⋊D4), C2.3(C5⋊D16), C4.1(D4.D5), C2.3(D8.D5), (C2×C40).75C22, C4.1(C23.D5), C20.60(C22⋊C4), C22.17(D4⋊D5), C2.6(D4⋊Dic5), C10.41(D4⋊C4), (C2×C52C16)⋊5C2, (C2×C4).119(C5⋊D4), SmallGroup(320,120)

Series: Derived Chief Lower central Upper central

C1C40 — C10.D16
C1C5C10C20C2×C20C2×C40C405C4 — C10.D16
C5C10C20C40 — C10.D16
C1C22C2×C4C2×C8C2×D8

Generators and relations for C10.D16
 G = < a,b,c | a10=b16=1, c2=a5, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 254 in 66 conjugacy classes, 31 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4, C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], C16, C4⋊C4, C2×C8, D8 [×2], D8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C2.D8, C2×C16, C2×D8, C40 [×2], C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C2.D16, C52C16, C4⋊Dic5, C2×C40, C5×D8 [×2], C5×D8, D4×C10, C2×C52C16, C405C4, C10×D8, C10.D16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, Dic5 [×2], D10, D4⋊C4, D16, SD32, C2×Dic5, C5⋊D4 [×2], C2.D16, D4⋊D5, D4.D5, C23.D5, C5⋊D16, D8.D5, D4⋊Dic5, C10.D16

Smallest permutation representation of C10.D16
On 160 points
Generators in S160
(1 125 91 65 106 150 53 43 23 141)(2 142 24 44 54 151 107 66 92 126)(3 127 93 67 108 152 55 45 25 143)(4 144 26 46 56 153 109 68 94 128)(5 113 95 69 110 154 57 47 27 129)(6 130 28 48 58 155 111 70 96 114)(7 115 81 71 112 156 59 33 29 131)(8 132 30 34 60 157 97 72 82 116)(9 117 83 73 98 158 61 35 31 133)(10 134 32 36 62 159 99 74 84 118)(11 119 85 75 100 160 63 37 17 135)(12 136 18 38 64 145 101 76 86 120)(13 121 87 77 102 146 49 39 19 137)(14 138 20 40 50 147 103 78 88 122)(15 123 89 79 104 148 51 41 21 139)(16 140 22 42 52 149 105 80 90 124)
(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 16 150 149)(2 148 151 15)(3 14 152 147)(4 146 153 13)(5 12 154 145)(6 160 155 11)(7 10 156 159)(8 158 157 9)(17 28 75 70)(18 69 76 27)(19 26 77 68)(20 67 78 25)(21 24 79 66)(22 65 80 23)(29 32 71 74)(30 73 72 31)(33 36 81 84)(34 83 82 35)(37 48 85 96)(38 95 86 47)(39 46 87 94)(40 93 88 45)(41 44 89 92)(42 91 90 43)(49 56 121 128)(50 127 122 55)(51 54 123 126)(52 125 124 53)(57 64 113 120)(58 119 114 63)(59 62 115 118)(60 117 116 61)(97 133 132 98)(99 131 134 112)(100 111 135 130)(101 129 136 110)(102 109 137 144)(103 143 138 108)(104 107 139 142)(105 141 140 106)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F10G···10N16A···16H20A20B20C20D40A···40H
order122222444455888810···1010···1016···162020202040···40
size1111882240402222222···28···810···1044444···4

50 irreducible representations

dim11111222222222224444
type+++++++++-+-++-
imageC1C2C2C2C4D4D4D5SD16D8D10Dic5D16SD32C5⋊D4C5⋊D4D4.D5D4⋊D5C5⋊D16D8.D5
kernelC10.D16C2×C52C16C405C4C10×D8C5×D8C40C2×C20C2×D8C20C2×C10C2×C8D8C10C10C8C2×C4C4C22C2C2
# reps11114112222444442244

Matrix representation of C10.D16 in GL4(𝔽241) generated by

143000
10215000
002400
000240
,
18517100
1245600
0017982
0020097
,
567000
11718500
009735
0041144
G:=sub<GL(4,GF(241))| [143,102,0,0,0,150,0,0,0,0,240,0,0,0,0,240],[185,124,0,0,171,56,0,0,0,0,179,200,0,0,82,97],[56,117,0,0,70,185,0,0,0,0,97,41,0,0,35,144] >;

C10.D16 in GAP, Magma, Sage, TeX

C_{10}.D_{16}
% in TeX

G:=Group("C10.D16");
// GroupNames label

G:=SmallGroup(320,120);
// by ID

G=gap.SmallGroup(320,120);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,675,346,192,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^10=b^16=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;
// generators/relations

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