Copied to
clipboard

## G = C52.4C4order 208 = 24·13

### 1st non-split extension by C52 of C4 acting via C4/C2=C2

Aliases: C52.4C4, C4.Dic13, C4.15D26, C134M4(2), C22.Dic13, C52.15C22, C132C85C2, (C2×C26).5C4, (C2×C52).5C2, (C2×C4).2D13, C26.14(C2×C4), C2.3(C2×Dic13), SmallGroup(208,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C52.4C4
 Chief series C1 — C13 — C26 — C52 — C13⋊2C8 — C52.4C4
 Lower central C13 — C26 — C52.4C4
 Upper central C1 — C4 — C2×C4

Generators and relations for C52.4C4
G = < a,b | a52=1, b4=a26, bab-1=a-1 >

Smallest permutation representation of C52.4C4
On 104 points
Generators in S104
```(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)
(1 95 14 82 27 69 40 56)(2 94 15 81 28 68 41 55)(3 93 16 80 29 67 42 54)(4 92 17 79 30 66 43 53)(5 91 18 78 31 65 44 104)(6 90 19 77 32 64 45 103)(7 89 20 76 33 63 46 102)(8 88 21 75 34 62 47 101)(9 87 22 74 35 61 48 100)(10 86 23 73 36 60 49 99)(11 85 24 72 37 59 50 98)(12 84 25 71 38 58 51 97)(13 83 26 70 39 57 52 96)```

`G:=sub<Sym(104)| (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), (1,95,14,82,27,69,40,56)(2,94,15,81,28,68,41,55)(3,93,16,80,29,67,42,54)(4,92,17,79,30,66,43,53)(5,91,18,78,31,65,44,104)(6,90,19,77,32,64,45,103)(7,89,20,76,33,63,46,102)(8,88,21,75,34,62,47,101)(9,87,22,74,35,61,48,100)(10,86,23,73,36,60,49,99)(11,85,24,72,37,59,50,98)(12,84,25,71,38,58,51,97)(13,83,26,70,39,57,52,96)>;`

`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), (1,95,14,82,27,69,40,56)(2,94,15,81,28,68,41,55)(3,93,16,80,29,67,42,54)(4,92,17,79,30,66,43,53)(5,91,18,78,31,65,44,104)(6,90,19,77,32,64,45,103)(7,89,20,76,33,63,46,102)(8,88,21,75,34,62,47,101)(9,87,22,74,35,61,48,100)(10,86,23,73,36,60,49,99)(11,85,24,72,37,59,50,98)(12,84,25,71,38,58,51,97)(13,83,26,70,39,57,52,96) );`

`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)], [(1,95,14,82,27,69,40,56),(2,94,15,81,28,68,41,55),(3,93,16,80,29,67,42,54),(4,92,17,79,30,66,43,53),(5,91,18,78,31,65,44,104),(6,90,19,77,32,64,45,103),(7,89,20,76,33,63,46,102),(8,88,21,75,34,62,47,101),(9,87,22,74,35,61,48,100),(10,86,23,73,36,60,49,99),(11,85,24,72,37,59,50,98),(12,84,25,71,38,58,51,97),(13,83,26,70,39,57,52,96)])`

C52.4C4 is a maximal subgroup of
D524C4  C104.6C4  C52.53D4  C52.46D4  C4.12D52  C52.D4  C52.10D4  C52.56D4  D52.3C4  M4(2)×D13  D526C22  Q8.D26  D4.Dic13  D4⋊D26  D4.9D26
C52.4C4 is a maximal quotient of
C26.7C42  C523C8  C52.55D4

58 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52X order 1 2 2 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 2 1 1 2 26 26 26 26 2 ··· 2 2 ··· 2 2 ··· 2

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 M4(2) D13 Dic13 D26 Dic13 C52.4C4 kernel C52.4C4 C13⋊2C8 C2×C52 C52 C2×C26 C13 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 2 6 6 6 6 24

Matrix representation of C52.4C4 in GL2(𝔽313) generated by

 162 0 74 114
,
 32 30 205 281
`G:=sub<GL(2,GF(313))| [162,74,0,114],[32,205,30,281] >;`

C52.4C4 in GAP, Magma, Sage, TeX

`C_{52}._4C_4`
`% in TeX`

`G:=Group("C52.4C4");`
`// GroupNames label`

`G:=SmallGroup(208,10);`
`// by ID`

`G=gap.SmallGroup(208,10);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,42,4804]);`
`// Polycyclic`

`G:=Group<a,b|a^52=1,b^4=a^26,b*a*b^-1=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽