Copied to
clipboard

## G = C10×SD32order 320 = 26·5

### Direct product of C10 and SD32

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C10×SD32
 Chief series C1 — C2 — C4 — C8 — C40 — C5×Q16 — C5×SD32 — C10×SD32
 Lower central C1 — C2 — C4 — C8 — C10×SD32
 Upper central C1 — C2×C10 — C2×C20 — C2×C40 — C10×SD32

Generators and relations for C10×SD32
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 210 in 90 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C2×SD32, C80, C2×C40, C5×D8, C5×D8, C5×Q16, C5×Q16, D4×C10, Q8×C10, C2×C80, C5×SD32, C10×D8, C10×Q16, C10×SD32
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, SD32, C2×D8, C5×D4, C22×C10, C2×SD32, C5×D8, D4×C10, C5×SD32, C10×D8, C10×SD32

Smallest permutation representation of C10×SD32
On 160 points
Generators in S160
(1 151 127 39 25 63 72 83 140 112)(2 152 128 40 26 64 73 84 141 97)(3 153 113 41 27 49 74 85 142 98)(4 154 114 42 28 50 75 86 143 99)(5 155 115 43 29 51 76 87 144 100)(6 156 116 44 30 52 77 88 129 101)(7 157 117 45 31 53 78 89 130 102)(8 158 118 46 32 54 79 90 131 103)(9 159 119 47 17 55 80 91 132 104)(10 160 120 48 18 56 65 92 133 105)(11 145 121 33 19 57 66 93 134 106)(12 146 122 34 20 58 67 94 135 107)(13 147 123 35 21 59 68 95 136 108)(14 148 124 36 22 60 69 96 137 109)(15 149 125 37 23 61 70 81 138 110)(16 150 126 38 24 62 71 82 139 111)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(65 71)(66 78)(67 69)(68 76)(70 74)(73 79)(75 77)(81 85)(82 92)(84 90)(86 88)(87 95)(89 93)(94 96)(97 103)(98 110)(99 101)(100 108)(102 106)(105 111)(107 109)(113 125)(114 116)(115 123)(117 121)(118 128)(120 126)(122 124)(129 143)(130 134)(131 141)(133 139)(135 137)(136 144)(138 142)(145 157)(146 148)(147 155)(149 153)(150 160)(152 158)(154 156)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 16A ··· 16H 20A ··· 20H 20I ··· 20P 40A ··· 40P 80A ··· 80AF order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 8 8 2 2 8 8 1 1 1 1 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 D4 D8 D8 SD32 C5×D4 C5×D4 C5×D8 C5×D8 C5×SD32 kernel C10×SD32 C2×C80 C5×SD32 C10×D8 C10×Q16 C2×SD32 C2×C16 SD32 C2×D8 C2×Q16 C40 C2×C20 C20 C2×C10 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 1 2 2 8 4 4 8 8 32

Matrix representation of C10×SD32 in GL3(𝔽241) generated by

 240 0 0 0 87 0 0 0 87
,
 1 0 0 0 103 41 0 200 103
,
 240 0 0 0 1 0 0 0 240
G:=sub<GL(3,GF(241))| [240,0,0,0,87,0,0,0,87],[1,0,0,0,103,200,0,41,103],[240,0,0,0,1,0,0,0,240] >;

C10×SD32 in GAP, Magma, Sage, TeX

C_{10}\times {\rm SD}_{32}
% in TeX

G:=Group("C10xSD32");
// GroupNames label

G:=SmallGroup(320,1007);
// by ID

G=gap.SmallGroup(320,1007);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,4204,2111,242,10085,5052,124]);
// 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^7>;
// generators/relations

׿
×
𝔽