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## G = C42.23D4order 128 = 27

### 5th 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 — C42.23D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C42.12C4 — C42.23D4
 Lower central C1 — C2 — C2×C4 — C42.23D4
 Upper central C1 — C22 — C2×C42 — C42.23D4
 Jennings C1 — C22 — C22 — C2×C42 — C42.23D4

Generators and relations for C42.23D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a-1b, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a-1b2c3 >

Subgroups: 112 in 63 conjugacy classes, 32 normal (24 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×4], C23, C42 [×4], C2×C8 [×6], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×2], C8⋊C4, C22⋊C8 [×2], C22⋊C8 [×2], C4⋊C8 [×3], C2×C42, C42.12C4, C42.6C4 [×2], C42.23D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C8.C4 [×2], C4.9C42, C4.C42, M4(2)⋊4C4, C42.23D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + - image C1 C2 C2 C4 C4 C4 D4 D4 Q8 C8.C4 C4.9C42 M4(2)⋊4C4 kernel C42.23D4 C42.12C4 C42.6C4 C4×C8 C8⋊C4 C22⋊C8 C42 C22×C4 C22×C4 C4 C2 C2 # reps 1 1 2 4 4 4 2 1 1 8 2 2

Matrix representation of C42.23D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 9 0 0 0 0 13 16 0 0 0 0 0 0 1 9 0 0 0 0 13 16
,
 13 0 0 0 0 0 0 13 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
,
 15 0 0 0 0 0 5 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 4 0 0 0 0 0 0 4 0 0
,
 9 1 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 4 0 0 0 0 0 9 0 0 8 0 0 0 0 0 2 9 0 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,1,13,0,0,0,0,9,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[15,5,0,0,0,0,0,8,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,8,2,0,0,0,0,0,9,0,0,8,0,0,0,0,0,4,9,0,0] >;`

C42.23D4 in GAP, Magma, Sage, TeX

`C_4^2._{23}D_4`
`% in TeX`

`G:=Group("C4^2.23D4");`
`// GroupNames label`

`G:=SmallGroup(128,19);`
`// by ID`

`G=gap.SmallGroup(128,19);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924,102]);`
`// Polycyclic`

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

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