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## G = C42.278D4order 128 = 27

### 260th 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 — C42.278D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×C4⋊Q8 — C42.278D4
 Lower central C1 — C2 — C2×C4 — C42.278D4
 Upper central C1 — C22 — C2×C42 — C42.278D4
 Jennings C1 — C2 — C2 — C2×C4 — C42.278D4

Generators and relations for C42.278D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2c3 >

Subgroups: 412 in 202 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×9], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C42.12C4, D4⋊Q8 [×4], C22.D8 [×4], C4.4D8 [×2], C82Q8 [×2], C2×C4⋊Q8, C22.26C24, C42.278D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×D8, C2×C8.C22, C42.278D4

Smallest permutation representation of C42.278D4
On 64 points
Generators in S64
```(1 56 25 63)(2 49 26 64)(3 50 27 57)(4 51 28 58)(5 52 29 59)(6 53 30 60)(7 54 31 61)(8 55 32 62)(9 47 38 19)(10 48 39 20)(11 41 40 21)(12 42 33 22)(13 43 34 23)(14 44 35 24)(15 45 36 17)(16 46 37 18)
(1 16 5 12)(2 34 6 38)(3 10 7 14)(4 36 8 40)(9 26 13 30)(11 28 15 32)(17 55 21 51)(18 59 22 63)(19 49 23 53)(20 61 24 57)(25 37 29 33)(27 39 31 35)(41 58 45 62)(42 56 46 52)(43 60 47 64)(44 50 48 54)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 11)(10 35)(12 33)(13 15)(14 39)(16 37)(17 43)(19 41)(20 24)(21 47)(23 45)(27 31)(34 36)(38 40)(44 48)(49 51)(50 61)(52 59)(53 55)(54 57)(56 63)(58 64)(60 62)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D8 C8.C22 2- 1+4 kernel C42.278D4 C42.12C4 D4⋊Q8 C22.D8 C4.4D8 C8⋊2Q8 C2×C4⋊Q8 C22.26C24 C42 C22×C4 C2×C4 C4 C4 # reps 1 1 4 4 2 2 1 1 2 2 8 2 2

Matrix representation of C42.278D4 in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16 0 0 0
,
 3 14 0 0 0 0 3 3 0 0 0 0 0 0 15 2 15 15 0 0 15 15 2 15 0 0 2 2 2 15 0 0 15 2 2 2
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,15,15,2,15,0,0,2,15,2,2,0,0,15,2,2,2,0,0,15,15,15,2],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;`

C42.278D4 in GAP, Magma, Sage, TeX

`C_4^2._{278}D_4`
`% in TeX`

`G:=Group("C4^2.278D4");`
`// GroupNames label`

`G:=SmallGroup(128,1958);`
`// by ID`

`G=gap.SmallGroup(128,1958);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,4037,1027,124]);`
`// Polycyclic`

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

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