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## G = C42.308D4order 128 = 27

### 4th non-split extension by C42 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.308D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C2×C4×C8 — C42.308D4
 Lower central C1 — C2 — C2×C4 — C42.308D4
 Upper central C1 — C2×C4 — C2×C42 — C42.308D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.308D4

Generators and relations for C42.308D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 340 in 190 conjugacy classes, 92 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C422C2 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C2×C4×C8, C4×D8, C4×SD16 [×2], C4×Q16, C88D4 [×2], C87D4, C8.18D4, C42.78C22 [×2], C8.12D4, C8.5Q8, C23.36C23 [×2], C42.308D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C4○D8 [×2], C42.308D4

Smallest permutation representation of C42.308D4
On 64 points
Generators in S64
```(1 49 5 53)(2 50 6 54)(3 51 7 55)(4 52 8 56)(9 45 13 41)(10 46 14 42)(11 47 15 43)(12 48 16 44)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(1 39 31 14)(2 40 32 15)(3 33 25 16)(4 34 26 9)(5 35 27 10)(6 36 28 11)(7 37 29 12)(8 38 30 13)(17 61 44 51)(18 62 45 52)(19 63 46 53)(20 64 47 54)(21 57 48 55)(22 58 41 56)(23 59 42 49)(24 60 43 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)
(1 48 5 44)(2 43 6 47)(3 46 7 42)(4 41 8 45)(9 58 13 62)(10 61 14 57)(11 64 15 60)(12 59 16 63)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)(33 53 37 49)(34 56 38 52)(35 51 39 55)(36 54 40 50)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 8A ··· 8P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 8 8 1 1 1 1 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 C4○D8 C4○D8 kernel C42.308D4 C2×C4×C8 C4×D8 C4×SD16 C4×Q16 C8⋊8D4 C8⋊7D4 C8.18D4 C42.78C22 C8.12D4 C8.5Q8 C23.36C23 C42 C22×C4 C8 C4 C22 # reps 1 1 1 2 1 2 1 1 2 1 1 2 2 2 8 8 8

Matrix representation of C42.308D4 in GL4(𝔽17) generated by

 4 9 0 0 4 13 0 0 0 0 0 1 0 0 16 0
,
 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 1 15 0 0 1 16 0 0 0 0 5 12 0 0 5 5
,
 16 2 0 0 0 1 0 0 0 0 4 0 0 0 0 13
`G:=sub<GL(4,GF(17))| [4,4,0,0,9,13,0,0,0,0,0,16,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,2,1,0,0,0,0,4,0,0,0,0,13] >;`

C42.308D4 in GAP, Magma, Sage, TeX

`C_4^2._{308}D_4`
`% in TeX`

`G:=Group("C4^2.308D4");`
`// GroupNames label`

`G:=SmallGroup(128,1900);`
`// by ID`

`G=gap.SmallGroup(128,1900);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,184,80,4037,124]);`
`// Polycyclic`

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

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