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## G = C5×SD64order 320 = 26·5

### Direct product of C5 and SD64

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C5×SD64, C322C10, C1604C2, D16.C10, Q321C10, C20.40D8, C40.69D4, C10.16D16, C80.20C22, C8.6(C5×D4), C4.2(C5×D8), (C5×Q32)⋊5C2, C2.4(C5×D16), C16.3(C2×C10), (C5×D16).2C2, SmallGroup(320,177)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×SD64
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b15 >

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 20G 20H 32A ··· 32H 40A ··· 40H 80A ··· 80P 160A ··· 160AF order 1 2 2 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 16 16 16 16 20 20 20 20 20 20 20 20 32 ··· 32 40 ··· 40 80 ··· 80 160 ··· 160 size 1 1 16 2 16 1 1 1 1 2 2 1 1 1 1 16 16 16 16 2 2 2 2 2 2 2 2 16 16 16 16 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D8 D16 C5×D4 SD64 C5×D8 C5×D16 C5×SD64 kernel C5×SD64 C160 C5×D16 C5×Q32 SD64 C32 D16 Q32 C40 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 4 4 4 4 1 2 4 4 8 8 16 32

Matrix representation of C5×SD64 in GL2(𝔽641) generated by

 531 0 0 531
,
 474 82 559 474
,
 287 354 354 354
G:=sub<GL(2,GF(641))| [531,0,0,531],[474,559,82,474],[287,354,354,354] >;

C5×SD64 in GAP, Magma, Sage, TeX

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

G:=Group("C5xSD64");
// GroupNames label

G:=SmallGroup(320,177);
// by ID

G=gap.SmallGroup(320,177);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,1120,309,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic

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

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