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## G = C42.27D4order 128 = 27

### 9th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.27D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊C8 — C42.27D4
 Lower central C1 — C2 — C2×C4 — C42.27D4
 Upper central C1 — C22 — C2×C42 — C42.27D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.27D4

Generators and relations for C42.27D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 136 in 76 conjugacy classes, 42 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×3], C22 [×3], C22 [×2], C8 [×6], C2×C4 [×4], C2×C4 [×6], C2×C4 [×3], C23, C42 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C4⋊C8 [×4], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C2×C4⋊C8, C4⋊M4(2) [×2], C42.27D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, C4.D4, C4.10D4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C4.10C42, C22.4Q16, C22.C42, C42.27D4

Smallest permutation representation of C42.27D4
On 64 points
Generators in S64
```(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)
(1 61 36 51)(2 52 37 62)(3 63 38 53)(4 54 39 64)(5 57 40 55)(6 56 33 58)(7 59 34 49)(8 50 35 60)(9 45 24 26)(10 27 17 46)(11 47 18 28)(12 29 19 48)(13 41 20 30)(14 31 21 42)(15 43 22 32)(16 25 23 44)
(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)
(1 31 3 29 5 27 7 25)(2 22 8 24 6 18 4 20)(9 33 11 39 13 37 15 35)(10 59 16 61 14 63 12 57)(17 49 23 51 21 53 19 55)(26 58 28 64 30 62 32 60)(34 44 36 42 38 48 40 46)(41 52 43 50 45 56 47 54)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + - + - + - image C1 C2 C2 C4 C4 D4 D4 Q8 D8 SD16 Q16 C4.D4 C4.10D4 C4.10C42 kernel C42.27D4 C2×C4⋊C8 C4⋊M4(2) C4⋊C8 C22×C8 C42 C22×C4 C22×C4 C2×C4 C2×C4 C2×C4 C4 C4 C2 # reps 1 1 2 8 4 2 1 1 2 4 2 1 1 2

Matrix representation of C42.27D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 16 1 0 0 0 0 15 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 8 16 0 0 0 0 14 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 9 0 0 0 15 0 0 0 0 0 0 2 0 0
,
 2 11 0 0 0 0 9 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,15,0,0,0,0,1,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,14,0,0,0,0,16,9,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,2,0,0,0,9,0,0,0,0,9,0,0,0],[2,9,0,0,0,0,11,15,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C42.27D4 in GAP, Magma, Sage, TeX

`C_4^2._{27}D_4`
`% in TeX`

`G:=Group("C4^2.27D4");`
`// GroupNames label`

`G:=SmallGroup(128,24);`
`// by ID`

`G=gap.SmallGroup(128,24);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,248,3924,242]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;`
`// generators/relations`

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