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## G = C10×M5(2)  order 320 = 26·5

### Direct product of C10 and M5(2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C10×M5(2)
 Chief series C1 — C2 — C4 — C8 — C40 — C80 — C5×M5(2) — C10×M5(2)
 Lower central C1 — C2 — C10×M5(2)
 Upper central C1 — C2×C40 — C10×M5(2)

Generators and relations for C10×M5(2)
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Subgroups: 98 in 90 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C10, C16, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, M5(2), C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, C2×M5(2), C80, C2×C40, C2×C40, C22×C20, C2×C80, C5×M5(2), C22×C40, C10×M5(2)
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, C22×C4, C20, C2×C10, M5(2), C22×C8, C40, C2×C20, C22×C10, C2×M5(2), C2×C40, C22×C20, C5×M5(2), C22×C40, C10×M5(2)

Smallest permutation representation of C10×M5(2)
On 160 points
Generators in S160
(1 49 82 125 71 157 29 97 34 134)(2 50 83 126 72 158 30 98 35 135)(3 51 84 127 73 159 31 99 36 136)(4 52 85 128 74 160 32 100 37 137)(5 53 86 113 75 145 17 101 38 138)(6 54 87 114 76 146 18 102 39 139)(7 55 88 115 77 147 19 103 40 140)(8 56 89 116 78 148 20 104 41 141)(9 57 90 117 79 149 21 105 42 142)(10 58 91 118 80 150 22 106 43 143)(11 59 92 119 65 151 23 107 44 144)(12 60 93 120 66 152 24 108 45 129)(13 61 94 121 67 153 25 109 46 130)(14 62 95 122 68 154 26 110 47 131)(15 63 96 123 69 155 27 111 48 132)(16 64 81 124 70 156 28 112 33 133)
(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)
(1 149)(2 158)(3 151)(4 160)(5 153)(6 146)(7 155)(8 148)(9 157)(10 150)(11 159)(12 152)(13 145)(14 154)(15 147)(16 156)(17 61)(18 54)(19 63)(20 56)(21 49)(22 58)(23 51)(24 60)(25 53)(26 62)(27 55)(28 64)(29 57)(30 50)(31 59)(32 52)(33 124)(34 117)(35 126)(36 119)(37 128)(38 121)(39 114)(40 123)(41 116)(42 125)(43 118)(44 127)(45 120)(46 113)(47 122)(48 115)(65 136)(66 129)(67 138)(68 131)(69 140)(70 133)(71 142)(72 135)(73 144)(74 137)(75 130)(76 139)(77 132)(78 141)(79 134)(80 143)(81 112)(82 105)(83 98)(84 107)(85 100)(86 109)(87 102)(88 111)(89 104)(90 97)(91 106)(92 99)(93 108)(94 101)(95 110)(96 103)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A ··· 8H 8I 8J 8K 8L 10A ··· 10L 10M ··· 10T 16A ··· 16P 20A ··· 20P 20Q ··· 20X 40A ··· 40AF 40AG ··· 40AV 80A ··· 80BL order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 8 8 8 10 ··· 10 10 ··· 10 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C5 C8 C8 C10 C10 C10 C20 C20 C40 C40 M5(2) C5×M5(2) kernel C10×M5(2) C2×C80 C5×M5(2) C22×C40 C2×C40 C22×C20 C2×M5(2) C2×C20 C22×C10 C2×C16 M5(2) C22×C8 C2×C8 C22×C4 C2×C4 C23 C10 C2 # reps 1 2 4 1 6 2 4 12 4 8 16 4 24 8 48 16 8 32

Matrix representation of C10×M5(2) in GL3(𝔽241) generated by

 240 0 0 0 150 0 0 0 150
,
 240 0 0 0 0 1 0 8 0
,
 1 0 0 0 1 0 0 0 240
G:=sub<GL(3,GF(241))| [240,0,0,0,150,0,0,0,150],[240,0,0,0,0,8,0,1,0],[1,0,0,0,1,0,0,0,240] >;

C10×M5(2) in GAP, Magma, Sage, TeX

C_{10}\times M_5(2)
% in TeX

G:=Group("C10xM5(2)");
// GroupNames label

G:=SmallGroup(320,1004);
// by ID

G=gap.SmallGroup(320,1004);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,2269,102,124]);
// Polycyclic

G:=Group<a,b,c|a^10=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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