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## G = C5⋊SD64order 320 = 26·5

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

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

 Derived series C1 — C80 — C5⋊SD64
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — D80 — C5⋊SD64
 Lower central C5 — C10 — C20 — C40 — C80 — C5⋊SD64
 Upper central C1 — C2 — C4 — C8 — C16 — Q32

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

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 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 32A ··· 32H 40A 40B 40C 40D 80A ··· 80H order 1 2 2 4 4 5 5 8 8 10 10 16 16 16 16 20 20 20 20 20 20 32 ··· 32 40 40 40 40 80 ··· 80 size 1 1 80 2 16 2 2 2 2 2 2 2 2 2 2 4 4 16 16 16 16 10 ··· 10 4 4 4 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 D8 D10 D16 C5⋊D4 SD64 D4⋊D5 C5⋊D16 C5⋊SD64 kernel C5⋊SD64 C5⋊2C32 D80 C5×Q32 C40 Q32 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 4 8 2 4 8

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

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 640 362
,
 37 312 0 0 329 37 0 0 0 0 419 17 0 0 419 222
,
 1 0 0 0 0 640 0 0 0 0 362 640 0 0 279 279
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

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