Copied to
clipboard

## G = C10.6M5(2)  order 320 = 26·5

### 3rd non-split extension by C10 of M5(2) acting via M5(2)/C2×C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10.6M5(2)
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C2×C5⋊C16 — C10.6M5(2)
 Lower central C5 — C10 — C10.6M5(2)
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C10.6M5(2)
G = < a,b,c | a10=b16=c2=1, bab-1=a3, ac=ca, cbc=a5b9 >

Subgroups: 162 in 66 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, C22×C8, C52C8, C52C8, C2×C20, C2×C20, C22×C10, C22⋊C16, C5⋊C16, C2×C52C8, C2×C52C8, C22×C20, C2×C5⋊C16, C22×C52C8, C10.6M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C16, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×C16, M5(2), C5⋊C8, C2×F5, C22⋊C16, C5⋊C16, C2×C5⋊C8, C22.F5, C22⋊F5, C2×C5⋊C16, C20.C8, C23.2F5, C10.6M5(2)

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 8I 8J 8K 8L 10A ··· 10G 16A ··· 16P 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 2 4 5 ··· 5 10 10 10 10 4 ··· 4 10 ··· 10 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + - - + image C1 C2 C2 C4 C4 C8 C8 C16 D4 M4(2) M5(2) F5 C5⋊C8 C2×F5 C5⋊C8 C22.F5 C22⋊F5 C5⋊C16 C20.C8 kernel C10.6M5(2) C2×C5⋊C16 C22×C5⋊2C8 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C2×C10 C5⋊2C8 C20 C10 C22×C4 C2×C4 C2×C4 C23 C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 2 2 4 1 1 1 1 2 2 4 4

Matrix representation of C10.6M5(2) in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 51 0 0 0 0 189 52 0 0 0 0 196 45 0 240 0 0 115 160 1 190
,
 74 4 0 0 0 0 137 167 0 0 0 0 0 0 0 0 240 1 0 0 81 126 239 51 0 0 230 169 115 0 0 0 64 53 115 0
,
 1 0 0 0 0 0 204 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,189,196,115,0,0,51,52,45,160,0,0,0,0,0,1,0,0,0,0,240,190],[74,137,0,0,0,0,4,167,0,0,0,0,0,0,0,81,230,64,0,0,0,126,169,53,0,0,240,239,115,115,0,0,1,51,0,0],[1,204,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C10.6M5(2) in GAP, Magma, Sage, TeX

`C_{10}._6M_5(2)`
`% in TeX`

`G:=Group("C10.6M5(2)");`
`// GroupNames label`

`G:=SmallGroup(320,249);`
`// by ID`

`G=gap.SmallGroup(320,249);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^16=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=a^5*b^9>;`
`// generators/relations`

׿
×
𝔽