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## G = C42.6C8order 128 = 27

### 3rd non-split extension by C42 of C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.6C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C4×C8 — C2×C4×C8 — C42.6C8
 Lower central C1 — C22 — C42.6C8
 Upper central C1 — C2×C8 — C42.6C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C42.6C8

Generators and relations for C42.6C8
G = < a,b,c | a4=b4=1, c8=a2, ab=ba, cac-1=a-1b2, cbc-1=a2b >

Subgroups: 100 in 76 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C2×C8, C2×C8, C22×C4, C4×C8, C2×C16, C2×C42, C22×C8, C165C4, C22⋊C16, C4⋊C16, C2×C4×C8, C42.6C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, M5(2), C42⋊C2, C22×C8, C2×M4(2), C42.12C4, C2×M5(2), C42.6C8

Smallest permutation representation of C42.6C8
On 64 points
Generators in S64
```(1 5 9 13)(2 18 10 26)(3 7 11 15)(4 20 12 28)(6 22 14 30)(8 24 16 32)(17 21 25 29)(19 23 27 31)(33 50 41 58)(34 38 42 46)(35 52 43 60)(36 40 44 48)(37 54 45 62)(39 56 47 64)(49 53 57 61)(51 55 59 63)
(1 38 21 59)(2 47 22 52)(3 40 23 61)(4 33 24 54)(5 42 25 63)(6 35 26 56)(7 44 27 49)(8 37 28 58)(9 46 29 51)(10 39 30 60)(11 48 31 53)(12 41 32 62)(13 34 17 55)(14 43 18 64)(15 36 19 57)(16 45 20 50)
(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)```

`G:=sub<Sym(64)| (1,5,9,13)(2,18,10,26)(3,7,11,15)(4,20,12,28)(6,22,14,30)(8,24,16,32)(17,21,25,29)(19,23,27,31)(33,50,41,58)(34,38,42,46)(35,52,43,60)(36,40,44,48)(37,54,45,62)(39,56,47,64)(49,53,57,61)(51,55,59,63), (1,38,21,59)(2,47,22,52)(3,40,23,61)(4,33,24,54)(5,42,25,63)(6,35,26,56)(7,44,27,49)(8,37,28,58)(9,46,29,51)(10,39,30,60)(11,48,31,53)(12,41,32,62)(13,34,17,55)(14,43,18,64)(15,36,19,57)(16,45,20,50), (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)>;`

`G:=Group( (1,5,9,13)(2,18,10,26)(3,7,11,15)(4,20,12,28)(6,22,14,30)(8,24,16,32)(17,21,25,29)(19,23,27,31)(33,50,41,58)(34,38,42,46)(35,52,43,60)(36,40,44,48)(37,54,45,62)(39,56,47,64)(49,53,57,61)(51,55,59,63), (1,38,21,59)(2,47,22,52)(3,40,23,61)(4,33,24,54)(5,42,25,63)(6,35,26,56)(7,44,27,49)(8,37,28,58)(9,46,29,51)(10,39,30,60)(11,48,31,53)(12,41,32,62)(13,34,17,55)(14,43,18,64)(15,36,19,57)(16,45,20,50), (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) );`

`G=PermutationGroup([[(1,5,9,13),(2,18,10,26),(3,7,11,15),(4,20,12,28),(6,22,14,30),(8,24,16,32),(17,21,25,29),(19,23,27,31),(33,50,41,58),(34,38,42,46),(35,52,43,60),(36,40,44,48),(37,54,45,62),(39,56,47,64),(49,53,57,61),(51,55,59,63)], [(1,38,21,59),(2,47,22,52),(3,40,23,61),(4,33,24,54),(5,42,25,63),(6,35,26,56),(7,44,27,49),(8,37,28,58),(9,46,29,51),(10,39,30,60),(11,48,31,53),(12,41,32,62),(13,34,17,55),(14,43,18,64),(15,36,19,57),(16,45,20,50)], [(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)]])`

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8H 8I ··· 8T 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 M4(2) C4○D4 M5(2) M5(2) kernel C42.6C8 C16⋊5C4 C22⋊C16 C4⋊C16 C2×C4×C8 C4×C8 C2×C42 C22×C8 C42 C22×C4 C8 C8 C4 C22 # reps 1 2 2 2 1 4 2 2 8 8 4 4 8 8

Matrix representation of C42.6C8 in GL4(𝔽17) generated by

 4 0 0 0 1 13 0 0 0 0 4 0 0 0 0 4
,
 16 0 0 0 4 1 0 0 0 0 0 4 0 0 4 0
,
 9 13 0 0 8 8 0 0 0 0 2 6 0 0 11 15
`G:=sub<GL(4,GF(17))| [4,1,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[16,4,0,0,0,1,0,0,0,0,0,4,0,0,4,0],[9,8,0,0,13,8,0,0,0,0,2,11,0,0,6,15] >;`

C42.6C8 in GAP, Magma, Sage, TeX

`C_4^2._6C_8`
`% in TeX`

`G:=Group("C4^2.6C8");`
`// GroupNames label`

`G:=SmallGroup(128,895);`
`// by ID`

`G=gap.SmallGroup(128,895);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,58,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=1,c^8=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^2*b>;`
`// generators/relations`

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