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G = D102SD16order 320 = 26·5

2nd semidirect product of D10 and SD16 acting via SD16/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102SD16, C52C821D4, C52(C88D4), C4⋊C4.24D10, Q8⋊C42D5, C4.166(D4×D5), D103Q81C2, C4⋊D20.3C2, C20.121(C2×D4), (C2×C8).210D10, (C2×Q8).16D10, D205C424C2, C2.18(D5×SD16), C4.32(C4○D20), C20.19(C4○D4), C10.69(C4○D8), C20.Q811C2, C10.32(C2×SD16), (C22×D5).81D4, C22.197(D4×D5), C2.8(Q8.D10), C10.23(C4⋊D4), (C2×C20).247C23, (C2×C40).200C22, (C2×Dic5).139D4, (C2×D20).67C22, C4⋊Dic5.94C22, (Q8×C10).30C22, C2.26(D10⋊D4), (D5×C2×C8)⋊21C2, (C2×Q8⋊D5)⋊3C2, (C5×Q8⋊C4)⋊22C2, (C2×C10).260(C2×D4), (C5×C4⋊C4).48C22, (C2×C4×D5).298C22, (C2×C4).354(C22×D5), (C2×C52C8).228C22, SmallGroup(320,434)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D102SD16
C1C5C10C20C2×C20C2×C4×D5D5×C2×C8 — D102SD16
C5C10C2×C20 — D102SD16
C1C22C2×C4Q8⋊C4

Generators and relations for D102SD16
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c3 >

Subgroups: 582 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×3], C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8 [×3], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8 [×2], C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5, C88D4, C8×D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4 [×2], Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C20.Q8, D205C4, C5×Q8⋊C4, C4⋊D20, D5×C2×C8, C2×Q8⋊D5, D103Q8, D102SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, C22×D5, C88D4, C4○D20, D4×D5 [×2], D10⋊D4, D5×SD16, Q8.D10, D102SD16

Smallest permutation representation of D102SD16
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)(11 151)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 130)(28 129)(29 128)(30 127)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 60)(41 133)(42 132)(43 131)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(61 114)(62 113)(63 112)(64 111)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 104)(72 103)(73 102)(74 101)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 93)(82 92)(83 91)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)
(1 50 15 68 55 78 28 95)(2 49 16 67 56 77 29 94)(3 48 17 66 57 76 30 93)(4 47 18 65 58 75 21 92)(5 46 19 64 59 74 22 91)(6 45 20 63 60 73 23 100)(7 44 11 62 51 72 24 99)(8 43 12 61 52 71 25 98)(9 42 13 70 53 80 26 97)(10 41 14 69 54 79 27 96)(31 103 125 85 143 140 153 113)(32 102 126 84 144 139 154 112)(33 101 127 83 145 138 155 111)(34 110 128 82 146 137 156 120)(35 109 129 81 147 136 157 119)(36 108 130 90 148 135 158 118)(37 107 121 89 149 134 159 117)(38 106 122 88 150 133 160 116)(39 105 123 87 141 132 151 115)(40 104 124 86 142 131 152 114)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 60)(7 59)(8 58)(9 57)(10 56)(11 19)(12 18)(13 17)(14 16)(21 25)(22 24)(26 30)(27 29)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 150)(40 149)(41 94)(42 93)(43 92)(44 91)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 140)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 124)(122 123)(125 130)(126 129)(127 128)(151 160)(152 159)(153 158)(154 157)(155 156)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444558888888810···102020202020···2040···40
size11111010402288101040222222101010102···244448···84···4

50 irreducible representations

dim11111111222222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10D10C4○D8C4○D20D4×D5D4×D5D5×SD16Q8.D10
kernelD102SD16C20.Q8D205C4C5×Q8⋊C4C4⋊D20D5×C2×C8C2×Q8⋊D5D103Q8C52C8C2×Dic5C22×D5Q8⋊C4C20D10C4⋊C4C2×C8C2×Q8C10C4C4C22C2C2
# reps11111111211224222482244

Matrix representation of D102SD16 in GL4(𝔽41) generated by

7600
34000
00400
00040
,
27200
51400
0009
00320
,
32100
213800
002626
001526
,
343500
8700
0010
00040
G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,40,0,0,0,0,40],[27,5,0,0,2,14,0,0,0,0,0,32,0,0,9,0],[3,21,0,0,21,38,0,0,0,0,26,15,0,0,26,26],[34,8,0,0,35,7,0,0,0,0,1,0,0,0,0,40] >;

D102SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("D10:2SD16");
// GroupNames label

G:=SmallGroup(320,434);
// by ID

G=gap.SmallGroup(320,434);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,438,102,12550]);
// Polycyclic

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

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