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G = C5⋊SD64order 320 = 26·5

The semidirect product of C5 and SD64 acting via SD64/Q32=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C53SD64, Q321D5, C20.7D8, D80.2C2, C16.6D10, C40.11D4, C10.10D16, C80.4C22, C52C323C2, (C5×Q32)⋊1C2, C4.3(D4⋊D5), C8.11(C5⋊D4), C2.6(C5⋊D16), SmallGroup(320,79)

Series: Derived Chief Lower central Upper central

C1C80 — C5⋊SD64
C1C5C10C20C40C80D80 — C5⋊SD64
C5C10C20C40C80 — C5⋊SD64
C1C2C4C8C16Q32

Generators and relations for C5⋊SD64
 G = < a,b,c | a5=b32=c2=1, bab-1=cac=a-1, cbc=b15 >

80C2
8C4
40C22
16D5
4Q8
20D4
8D10
8C20
2Q16
10D8
4D20
4C5×Q8
5C32
5D16
2D40
2C5×Q16
5SD64

Smallest permutation representation of C5⋊SD64
On 160 points
Generators in S160
(1 63 145 87 112)(2 113 88 146 64)(3 33 147 89 114)(4 115 90 148 34)(5 35 149 91 116)(6 117 92 150 36)(7 37 151 93 118)(8 119 94 152 38)(9 39 153 95 120)(10 121 96 154 40)(11 41 155 65 122)(12 123 66 156 42)(13 43 157 67 124)(14 125 68 158 44)(15 45 159 69 126)(16 127 70 160 46)(17 47 129 71 128)(18 97 72 130 48)(19 49 131 73 98)(20 99 74 132 50)(21 51 133 75 100)(22 101 76 134 52)(23 53 135 77 102)(24 103 78 136 54)(25 55 137 79 104)(26 105 80 138 56)(27 57 139 81 106)(28 107 82 140 58)(29 59 141 83 108)(30 109 84 142 60)(31 61 143 85 110)(32 111 86 144 62)
(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)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 110)(34 125)(35 108)(36 123)(37 106)(38 121)(39 104)(40 119)(41 102)(42 117)(43 100)(44 115)(45 98)(46 113)(47 128)(48 111)(49 126)(50 109)(51 124)(52 107)(53 122)(54 105)(55 120)(56 103)(57 118)(58 101)(59 116)(60 99)(61 114)(62 97)(63 112)(64 127)(65 135)(66 150)(67 133)(68 148)(69 131)(70 146)(71 129)(72 144)(73 159)(74 142)(75 157)(76 140)(77 155)(78 138)(79 153)(80 136)(81 151)(82 134)(83 149)(84 132)(85 147)(86 130)(87 145)(88 160)(89 143)(90 158)(91 141)(92 156)(93 139)(94 154)(95 137)(96 152)

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

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

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

41 conjugacy classes

class 1 2A2B4A4B5A5B8A8B10A10B16A16B16C16D20A20B20C20D20E20F32A···32H40A40B40C40D80A···80H
order12244558810101616161620202020202032···324040404080···80
size11802162222222222441616161610···1044444···4

41 irreducible representations

dim11112222222444
type++++++++++++
imageC1C2C2C2D4D5D8D10D16C5⋊D4SD64D4⋊D5C5⋊D16C5⋊SD64
kernelC5⋊SD64C52C32D80C5×Q32C40Q32C20C16C10C8C5C4C2C1
# reps11111222448248

Matrix representation of C5⋊SD64 in GL4(𝔽641) generated by

1000
0100
0001
00640362
,
3731200
3293700
0041917
00419222
,
1000
064000
00362640
00279279
G:=sub<GL(4,GF(641))| [1,0,0,0,0,1,0,0,0,0,0,640,0,0,1,362],[37,329,0,0,312,37,0,0,0,0,419,419,0,0,17,222],[1,0,0,0,0,640,0,0,0,0,362,279,0,0,640,279] >;

C5⋊SD64 in GAP, Magma, Sage, TeX

C_5\rtimes {\rm SD}_{64}
% in TeX

G:=Group("C5:SD64");
// GroupNames label

G:=SmallGroup(320,79);
// by ID

G=gap.SmallGroup(320,79);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,232,254,135,142,675,346,192,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^5=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^15>;
// generators/relations

Export

Subgroup lattice of C5⋊SD64 in TeX

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