Copied to
clipboard

## G = C5⋊C16.C22order 320 = 26·5

### The non-split extension by C5⋊C16 of C22 acting via C22/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5⋊C16.C22
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C5⋊C16.C22
 Lower central C5 — C10 — C5⋊C16.C22
 Upper central C1 — C4 — C4○D4

Generators and relations for C5⋊C16.C22
G = < a,b,c,d | a5=b16=c2=1, d2=b2, bab-1=dad-1=a3, ac=ca, cbc=b9, bd=db, cd=dc >

Subgroups: 178 in 84 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, D4, Q8, C10, C10, C16, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C2×C16, M5(2), C8○D4, C52C8, C52C8, C2×C20, C5×D4, C5×Q8, D4○C16, C5⋊C16, C5⋊C16, C2×C52C8, C4.Dic5, C5×C4○D4, C2×C5⋊C16, C20.C8, D4.Dic5, C5⋊C16.C22
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, C22×C8, C5⋊C8, C2×F5, D4○C16, C2×C5⋊C8, C22×F5, C22×C5⋊C8, C5⋊C16.C22

Smallest permutation representation of C5⋊C16.C22
On 160 points
Generators in S160
```(1 19 75 62 143)(2 63 20 144 76)(3 129 64 77 21)(4 78 130 22 49)(5 23 79 50 131)(6 51 24 132 80)(7 133 52 65 25)(8 66 134 26 53)(9 27 67 54 135)(10 55 28 136 68)(11 137 56 69 29)(12 70 138 30 57)(13 31 71 58 139)(14 59 32 140 72)(15 141 60 73 17)(16 74 142 18 61)(33 125 92 111 149)(34 112 126 150 93)(35 151 97 94 127)(36 95 152 128 98)(37 113 96 99 153)(38 100 114 154 81)(39 155 101 82 115)(40 83 156 116 102)(41 117 84 103 157)(42 104 118 158 85)(43 159 105 86 119)(44 87 160 120 106)(45 121 88 107 145)(46 108 122 146 89)(47 147 109 90 123)(48 91 148 124 110)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(113 121)(115 123)(117 125)(119 127)(130 138)(132 140)(134 142)(136 144)(145 153)(147 155)(149 157)(151 159)
(1 112 3 98 5 100 7 102 9 104 11 106 13 108 15 110)(2 97 4 99 6 101 8 103 10 105 12 107 14 109 16 111)(17 148 19 150 21 152 23 154 25 156 27 158 29 160 31 146)(18 149 20 151 22 153 24 155 26 157 28 159 30 145 32 147)(33 76 35 78 37 80 39 66 41 68 43 70 45 72 47 74)(34 77 36 79 38 65 40 67 42 69 44 71 46 73 48 75)(49 113 51 115 53 117 55 119 57 121 59 123 61 125 63 127)(50 114 52 116 54 118 56 120 58 122 60 124 62 126 64 128)(81 133 83 135 85 137 87 139 89 141 91 143 93 129 95 131)(82 134 84 136 86 138 88 140 90 142 92 144 94 130 96 132)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C 10D 16A ··· 16H 16I ··· 16T 20A 20B 20C 20D 20E order 1 2 2 2 2 4 4 4 4 4 5 8 8 8 8 8 ··· 8 10 10 10 10 16 ··· 16 16 ··· 16 20 20 20 20 20 size 1 1 2 2 2 1 1 2 2 2 4 5 5 5 5 10 ··· 10 4 8 8 8 5 ··· 5 10 ··· 10 4 4 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 8 type + + + + + + - - image C1 C2 C2 C2 C4 C4 C8 C8 D4○C16 F5 C2×F5 C5⋊C8 C5⋊C8 C5⋊C16.C22 kernel C5⋊C16.C22 C2×C5⋊C16 C20.C8 D4.Dic5 C4.Dic5 C5×C4○D4 C5×D4 C5×Q8 C5 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 1 6 2 12 4 8 1 3 3 1 2

Matrix representation of C5⋊C16.C22 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 0 165 0 0 0 0 165 0 0 0 0 0 0 0 79 218 106 81 0 0 129 104 23 102 0 0 160 239 137 25 0 0 139 27 2 162
,
 1 0 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
,
 165 0 0 0 0 0 0 165 0 0 0 0 0 0 79 218 106 81 0 0 129 104 23 102 0 0 160 239 137 25 0 0 139 27 2 162

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,165,0,0,0,0,165,0,0,0,0,0,0,0,79,129,160,139,0,0,218,104,239,27,0,0,106,23,137,2,0,0,81,102,25,162],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[165,0,0,0,0,0,0,165,0,0,0,0,0,0,79,129,160,139,0,0,218,104,239,27,0,0,106,23,137,2,0,0,81,102,25,162] >;`

C5⋊C16.C22 in GAP, Magma, Sage, TeX

`C_5\rtimes C_{16}.C_2^2`
`% in TeX`

`G:=Group("C5:C16.C2^2");`
`// GroupNames label`

`G:=SmallGroup(320,1129);`
`// by ID`

`G=gap.SmallGroup(320,1129);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,80,102,6278,1595]);`
`// Polycyclic`

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

׿
×
𝔽