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## G = C10×2- 1+4order 320 = 26·5

### Direct product of C10 and 2- 1+4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C10×2- 1+4
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — C5×C4○D4 — C5×2- 1+4 — C10×2- 1+4
 Lower central C1 — C2 — C10×2- 1+4
 Upper central C1 — C2×C10 — C10×2- 1+4

Generators and relations for C10×2- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 834 in 794 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C10, C2×C10, C2×C10, C22×Q8, C2×C4○D4, 2- 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, C2×2- 1+4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, Q8×C2×C10, C10×C4○D4, C5×2- 1+4, C10×2- 1+4
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2- 1+4, C25, C22×C10, C2×2- 1+4, C23×C10, C5×2- 1+4, C24×C10, C10×2- 1+4

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

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

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

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

170 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A ··· 4T 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AZ 20A ··· 20CB order 1 2 2 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + - image C1 C2 C2 C2 C5 C10 C10 C10 2- 1+4 C5×2- 1+4 kernel C10×2- 1+4 Q8×C2×C10 C10×C4○D4 C5×2- 1+4 C2×2- 1+4 C22×Q8 C2×C4○D4 2- 1+4 C10 C2 # reps 1 5 10 16 4 20 40 64 2 8

Matrix representation of C10×2- 1+4 in GL5(𝔽41)

 40 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31
,
 1 0 0 0 0 0 22 39 11 30 0 15 14 31 21 0 19 19 39 3 0 38 5 33 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 12 30 40 0 0 16 11 0 40
,
 1 0 0 0 0 0 27 39 0 0 0 37 14 0 0 0 28 19 0 40 0 24 5 1 0
,
 40 0 0 0 0 0 10 11 0 0 0 2 31 0 0 0 13 39 26 26 0 20 32 26 15

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[1,0,0,0,0,0,22,15,19,38,0,39,14,19,5,0,11,31,39,33,0,30,21,3,7],[1,0,0,0,0,0,1,0,12,16,0,0,1,30,11,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,27,37,28,24,0,39,14,19,5,0,0,0,0,1,0,0,0,40,0],[40,0,0,0,0,0,10,2,13,20,0,11,31,39,32,0,0,0,26,26,0,0,0,26,15] >;

C10×2- 1+4 in GAP, Magma, Sage, TeX

C_{10}\times 2_-^{1+4}
% in TeX

G:=Group("C10xES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1633);
// by ID

G=gap.SmallGroup(320,1633);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269,1128,1731,856,4707]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

׿
×
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