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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 [×2], C4, C4 [×4], C22 [×2], C5, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5, C10, C10, C16, C16, C2×C8, D8, SD16, Q16, Q16 [×3], C2×Q8, C4○D4, Dic5, Dic5 [×2], C20, C20, D10, C2×C10, M5(2), SD32, SD32, Q32 [×2], C2×Q16, C4○D8, C52C8, C40, Dic10 [×3], C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, Q32⋊C2, C52C16, C80, C8×D5, Dic20 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], 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 62)(2 53)(3 60)(4 51)(5 58)(6 49)(7 56)(8 63)(9 54)(10 61)(11 52)(12 59)(13 50)(14 57)(15 64)(16 55)(17 76)(18 67)(19 74)(20 65)(21 72)(22 79)(23 70)(24 77)(25 68)(26 75)(27 66)(28 73)(29 80)(30 71)(31 78)(32 69)(33 124)(34 115)(35 122)(36 113)(37 120)(38 127)(39 118)(40 125)(41 116)(42 123)(43 114)(44 121)(45 128)(46 119)(47 126)(48 117)(81 108)(82 99)(83 106)(84 97)(85 104)(86 111)(87 102)(88 109)(89 100)(90 107)(91 98)(92 105)(93 112)(94 103)(95 110)(96 101)(129 147)(130 154)(131 145)(132 152)(133 159)(134 150)(135 157)(136 148)(137 155)(138 146)(139 153)(140 160)(141 151)(142 158)(143 149)(144 156)
(1 147 23 95 35)(2 148 24 96 36)(3 149 25 81 37)(4 150 26 82 38)(5 151 27 83 39)(6 152 28 84 40)(7 153 29 85 41)(8 154 30 86 42)(9 155 31 87 43)(10 156 32 88 44)(11 157 17 89 45)(12 158 18 90 46)(13 159 19 91 47)(14 160 20 92 48)(15 145 21 93 33)(16 146 22 94 34)(49 132 73 97 125)(50 133 74 98 126)(51 134 75 99 127)(52 135 76 100 128)(53 136 77 101 113)(54 137 78 102 114)(55 138 79 103 115)(56 139 80 104 116)(57 140 65 105 117)(58 141 66 106 118)(59 142 67 107 119)(60 143 68 108 120)(61 144 69 109 121)(62 129 70 110 122)(63 130 71 111 123)(64 131 72 112 124)
(1 35)(2 44)(3 37)(4 46)(5 39)(6 48)(7 41)(8 34)(9 43)(10 36)(11 45)(12 38)(13 47)(14 40)(15 33)(16 42)(18 26)(20 28)(22 30)(24 32)(49 125)(50 118)(51 127)(52 120)(53 113)(54 122)(55 115)(56 124)(57 117)(58 126)(59 119)(60 128)(61 121)(62 114)(63 123)(64 116)(66 74)(68 76)(70 78)(72 80)(81 149)(82 158)(83 151)(84 160)(85 153)(86 146)(87 155)(88 148)(89 157)(90 150)(91 159)(92 152)(93 145)(94 154)(95 147)(96 156)(97 132)(98 141)(99 134)(100 143)(101 136)(102 129)(103 138)(104 131)(105 140)(106 133)(107 142)(108 135)(109 144)(110 137)(111 130)(112 139)

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

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

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

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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