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G = SD32⋊D5order 320 = 26·5

2nd semidirect product of SD32 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD322D5, D8.4D10, C16.2D10, Dic406C2, D10.16D8, C80.9C22, Q16.1D10, C40.18C23, Dic5.18D8, Dic20.3C22, C4.6(D4×D5), D8.D54C2, (D5×Q16)⋊4C2, C5⋊Q321C2, C80⋊C22C2, (C4×D5).9D4, C2.21(D5×D8), C52C8.4D4, (C5×SD32)⋊2C2, C10.37(C2×D8), C20.12(C2×D4), C52(Q32⋊C2), D83D5.1C2, (C8×D5).5C22, (C5×D8).4C22, C8.24(C22×D5), C52C16.1C22, (C5×Q16).2C22, SmallGroup(320,542)

Series: Derived Chief Lower central Upper central

C1C40 — SD32⋊D5
C1C5C10C20C40C8×D5D5×Q16 — SD32⋊D5
C5C10C20C40 — SD32⋊D5
C1C2C4C8SD32

Generators and relations for SD32⋊D5
 G = < a,b,c,d | a16=b2=c5=d2=1, bab=a7, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 374 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, M5(2), SD32, SD32, Q32, C2×Q16, C4○D8, C52C8, C40, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, Q32⋊C2, C52C16, C80, C8×D5, Dic20, D4.D5, C5⋊Q16, C5×D8, C5×Q16, D42D5, Q8×D5, C80⋊C2, Dic40, D8.D5, C5⋊Q32, C5×SD32, D83D5, D5×Q16, SD32⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, Q32⋊C2, D4×D5, D5×D8, SD32⋊D5

Smallest permutation representation of SD32⋊D5
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)
(1 139)(2 130)(3 137)(4 144)(5 135)(6 142)(7 133)(8 140)(9 131)(10 138)(11 129)(12 136)(13 143)(14 134)(15 141)(16 132)(17 89)(18 96)(19 87)(20 94)(21 85)(22 92)(23 83)(24 90)(25 81)(26 88)(27 95)(28 86)(29 93)(30 84)(31 91)(32 82)(33 147)(34 154)(35 145)(36 152)(37 159)(38 150)(39 157)(40 148)(41 155)(42 146)(43 153)(44 160)(45 151)(46 158)(47 149)(48 156)(49 111)(50 102)(51 109)(52 100)(53 107)(54 98)(55 105)(56 112)(57 103)(58 110)(59 101)(60 108)(61 99)(62 106)(63 97)(64 104)(65 115)(66 122)(67 113)(68 120)(69 127)(70 118)(71 125)(72 116)(73 123)(74 114)(75 121)(76 128)(77 119)(78 126)(79 117)(80 124)
(1 61 19 159 79)(2 62 20 160 80)(3 63 21 145 65)(4 64 22 146 66)(5 49 23 147 67)(6 50 24 148 68)(7 51 25 149 69)(8 52 26 150 70)(9 53 27 151 71)(10 54 28 152 72)(11 55 29 153 73)(12 56 30 154 74)(13 57 31 155 75)(14 58 32 156 76)(15 59 17 157 77)(16 60 18 158 78)(33 113 135 111 83)(34 114 136 112 84)(35 115 137 97 85)(36 116 138 98 86)(37 117 139 99 87)(38 118 140 100 88)(39 119 141 101 89)(40 120 142 102 90)(41 121 143 103 91)(42 122 144 104 92)(43 123 129 105 93)(44 124 130 106 94)(45 125 131 107 95)(46 126 132 108 96)(47 127 133 109 81)(48 128 134 110 82)
(1 79)(2 72)(3 65)(4 74)(5 67)(6 76)(7 69)(8 78)(9 71)(10 80)(11 73)(12 66)(13 75)(14 68)(15 77)(16 70)(18 26)(20 28)(22 30)(24 32)(33 103)(34 112)(35 105)(36 98)(37 107)(38 100)(39 109)(40 102)(41 111)(42 104)(43 97)(44 106)(45 99)(46 108)(47 101)(48 110)(49 147)(50 156)(51 149)(52 158)(53 151)(54 160)(55 153)(56 146)(57 155)(58 148)(59 157)(60 150)(61 159)(62 152)(63 145)(64 154)(81 89)(83 91)(85 93)(87 95)(113 143)(114 136)(115 129)(116 138)(117 131)(118 140)(119 133)(120 142)(121 135)(122 144)(123 137)(124 130)(125 139)(126 132)(127 141)(128 134)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C10A10B10C10D16A16B16C16D20A20B20C20D40A40B40C40D80A···80H
order122244444558881010101016161616202020204040404080···80
size118102810404022222022161644202044161644444···4

38 irreducible representations

dim11111111222222224444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D5D8D8D10D10D10Q32⋊C2D4×D5D5×D8SD32⋊D5
kernelSD32⋊D5C80⋊C2Dic40D8.D5C5⋊Q32C5×SD32D83D5D5×Q16C52C8C4×D5SD32Dic5D10C16D8Q16C5C4C2C1
# reps11111111112222222248

Matrix representation of SD32⋊D5 in GL4(𝔽241) generated by

32110168133
131209168168
10349228120
13810312113
,
20823380224
8338080
141392088
10014123333
,
189100
240000
0001
00240189
,
2405200
0100
000240
002400
G:=sub<GL(4,GF(241))| [32,131,103,138,110,209,49,103,168,168,228,121,133,168,120,13],[208,8,141,100,233,33,39,141,80,80,208,233,224,80,8,33],[189,240,0,0,1,0,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,52,1,0,0,0,0,0,240,0,0,240,0] >;

SD32⋊D5 in GAP, Magma, Sage, TeX

{\rm SD}_{32}\rtimes D_5
% in TeX

G:=Group("SD32:D5");
// GroupNames label

G:=SmallGroup(320,542);
// by ID

G=gap.SmallGroup(320,542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,135,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^7,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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