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G = D10.17SD16order 320 = 26·5

2nd non-split extension by D10 of SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.17SD16, C4.Q87D5, C4⋊C4.37D10, (C2×C8).138D10, C4⋊D20.5C2, D205C431C2, D206C414C2, D101C830C2, C2.23(D5×SD16), C20.28(C4○D4), C4.73(C4○D20), C20.Q817C2, (C2×Dic5).49D4, C10.39(C2×SD16), C22.215(D4×D5), C2.21(D40⋊C2), C10.69(C8⋊C22), (C2×C40).285C22, (C2×C20).279C23, C4.25(Q82D5), (C2×D20).77C22, (C22×D5).120D4, C53(C23.46D4), C4⋊Dic5.111C22, C2.12(D10.13D4), C10.42(C22.D4), (D5×C4⋊C4)⋊6C2, (C5×C4.Q8)⋊16C2, (C2×C4×D5).36C22, (C2×C10).284(C2×D4), (C5×C4⋊C4).72C22, (C2×C52C8).57C22, (C2×C4).382(C22×D5), SmallGroup(320,490)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.17SD16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.17SD16
C5C10C2×C20 — D10.17SD16
C1C22C2×C4C4.Q8

Generators and relations for D10.17SD16
 G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c3 >

Subgroups: 574 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C22⋊C8, D4⋊C4 [×2], C4.Q8, C4.Q8, C2×C4⋊C4, C4⋊D4, C52C8, C40, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C23.46D4, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C20.Q8, D206C4, D101C8, D205C4, C5×C4.Q8, D5×C4⋊C4, C4⋊D20, D10.17SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8⋊C22, C22×D5, C23.46D4, C4○D20, D4×D5, Q82D5, D10.13D4, D5×SD16, D40⋊C2, D10.17SD16

Smallest permutation representation of D10.17SD16
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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 150)(9 149)(10 148)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 140)(20 139)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(51 128)(52 127)(53 126)(54 125)(55 124)(56 123)(57 122)(58 121)(59 130)(60 129)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 110)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 120)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 100)(90 99)
(1 63 45 55 39 78 20 90)(2 64 46 56 40 79 11 81)(3 65 47 57 31 80 12 82)(4 66 48 58 32 71 13 83)(5 67 49 59 33 72 14 84)(6 68 50 60 34 73 15 85)(7 69 41 51 35 74 16 86)(8 70 42 52 36 75 17 87)(9 61 43 53 37 76 18 88)(10 62 44 54 38 77 19 89)(21 130 153 118 140 95 148 103)(22 121 154 119 131 96 149 104)(23 122 155 120 132 97 150 105)(24 123 156 111 133 98 141 106)(25 124 157 112 134 99 142 107)(26 125 158 113 135 100 143 108)(27 126 159 114 136 91 144 109)(28 127 160 115 137 92 145 110)(29 128 151 116 138 93 146 101)(30 129 152 117 139 94 147 102)
(1 135 6 140)(2 136 7 131)(3 137 8 132)(4 138 9 133)(5 139 10 134)(11 144 16 149)(12 145 17 150)(13 146 18 141)(14 147 19 142)(15 148 20 143)(21 39 26 34)(22 40 27 35)(23 31 28 36)(24 32 29 37)(25 33 30 38)(41 154 46 159)(42 155 47 160)(43 156 48 151)(44 157 49 152)(45 158 50 153)(51 96 56 91)(52 97 57 92)(53 98 58 93)(54 99 59 94)(55 100 60 95)(61 106 66 101)(62 107 67 102)(63 108 68 103)(64 109 69 104)(65 110 70 105)(71 116 76 111)(72 117 77 112)(73 118 78 113)(74 119 79 114)(75 120 80 115)(81 126 86 121)(82 127 87 122)(83 128 88 123)(84 129 89 124)(85 130 90 125)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111110104022448202020224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4SD16D10D10C4○D20C8⋊C22Q82D5D4×D5D5×SD16D40⋊C2
kernelD10.17SD16C20.Q8D206C4D101C8D205C4C5×C4.Q8D5×C4⋊C4C4⋊D20C2×Dic5C22×D5C4.Q8C20D10C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111111124442812244

Matrix representation of D10.17SD16 in GL4(𝔽41) generated by

1000
0100
00034
00635
,
1000
0100
002527
002716
,
262600
152600
001840
003623
,
04000
40000
0029
00439
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,6,0,0,34,35],[1,0,0,0,0,1,0,0,0,0,25,27,0,0,27,16],[26,15,0,0,26,26,0,0,0,0,18,36,0,0,40,23],[0,40,0,0,40,0,0,0,0,0,2,4,0,0,9,39] >;

D10.17SD16 in GAP, Magma, Sage, TeX

D_{10}._{17}{\rm SD}_{16}
% in TeX

G:=Group("D10.17SD16");
// GroupNames label

G:=SmallGroup(320,490);
// by ID

G=gap.SmallGroup(320,490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,100,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,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^3>;
// generators/relations

׿
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