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G = D108SD16order 320 = 26·5

2nd semidirect product of D10 and SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D108SD16, Dic107D4, (C5×Q8)⋊5D4, C4.63(D4×D5), Q82(C5⋊D4), C54(Q8⋊D4), C20.48(C2×D4), (C2×SD16)⋊11D5, (C2×D4).72D10, C202D4.8C2, (C2×C8).147D10, D101C833C2, C2.29(D5×SD16), C10.58C22≀C2, (C10×SD16)⋊21C2, Q8⋊Dic528C2, (C2×Q8).116D10, (C2×Dic5).80D4, C10.46(C2×SD16), C22.267(D4×D5), C20.44D435C2, (C2×C40).294C22, (C2×C20).447C23, (C22×D5).129D4, (D4×C10).96C22, (Q8×C10).76C22, C2.26(C23⋊D10), C2.29(SD16⋊D5), C10.49(C8.C22), C4⋊Dic5.174C22, (C2×Dic10).131C22, (C2×Q8×D5)⋊2C2, C4.43(C2×C5⋊D4), (C2×D4.D5)⋊20C2, (C2×C4×D5).52C22, (C2×C10).359(C2×D4), (C2×C4).536(C22×D5), (C2×C52C8).157C22, SmallGroup(320,797)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D108SD16
C1C5C10C2×C10C2×C20C2×C4×D5C2×Q8×D5 — D108SD16
C5C10C2×C20 — D108SD16
C1C22C2×C4C2×SD16

Generators and relations for D108SD16
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c3 >

Subgroups: 654 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×10], D4 [×4], Q8 [×2], Q8 [×8], C23 [×2], D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16, C2×SD16, C22×Q8, C52C8, C40, Dic10 [×2], Dic10 [×5], C4×D5 [×6], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, Q8⋊D4, C2×C52C8, C4⋊Dic5, D4.D5 [×2], C23.D5, C2×C40, C5×SD16 [×2], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D101C8, Q8⋊Dic5, C2×D4.D5, C202D4, C10×SD16, C2×Q8×D5, D108SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, C5⋊D4 [×2], C22×D5, Q8⋊D4, D4×D5 [×2], C2×C5⋊D4, D5×SD16, SD16⋊D5, C23⋊D10, D108SD16

Smallest permutation representation of D108SD16
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 26 50 153 33 148 14 140)(2 27 41 154 34 149 15 131)(3 28 42 155 35 150 16 132)(4 29 43 156 36 141 17 133)(5 30 44 157 37 142 18 134)(6 21 45 158 38 143 19 135)(7 22 46 159 39 144 20 136)(8 23 47 160 40 145 11 137)(9 24 48 151 31 146 12 138)(10 25 49 152 32 147 13 139)(51 109 69 114 74 121 81 91)(52 110 70 115 75 122 82 92)(53 101 61 116 76 123 83 93)(54 102 62 117 77 124 84 94)(55 103 63 118 78 125 85 95)(56 104 64 119 79 126 86 96)(57 105 65 120 80 127 87 97)(58 106 66 111 71 128 88 98)(59 107 67 112 72 129 89 99)(60 108 68 113 73 130 90 100)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 118)(22 119)(23 120)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(91 149)(92 150)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 156)(102 157)(103 158)(104 159)(105 160)(106 151)(107 152)(108 153)(109 154)(110 155)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444455888810···1010101010202020202020202040···40
size111181010224420202040224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D5SD16D10D10D10C5⋊D4C8.C22D4×D5D4×D5D5×SD16SD16⋊D5
kernelD108SD16C20.44D4D101C8Q8⋊Dic5C2×D4.D5C202D4C10×SD16C2×Q8×D5Dic10C2×Dic5C5×Q8C22×D5C2×SD16D10C2×C8C2×D4C2×Q8Q8C10C4C22C2C2
# reps11111111212124222812244

Matrix representation of D108SD16 in GL4(𝔽41) generated by

6600
35100
0010
0001
,
6600
13500
00400
00040
,
23600
351800
002615
002626
,
23600
351800
001938
003822
G:=sub<GL(4,GF(41))| [6,35,0,0,6,1,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,6,35,0,0,0,0,40,0,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,26,26,0,0,15,26],[23,35,0,0,6,18,0,0,0,0,19,38,0,0,38,22] >;

D108SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("D10:8SD16");
// GroupNames label

G:=SmallGroup(320,797);
// by ID

G=gap.SmallGroup(320,797);
# by ID

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

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

׿
×
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