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## G = C42.315D4order 128 = 27

### 11st 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 — C42.315D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C22.26C24 — C42.315D4
 Lower central C1 — C2 — C2×C4 — C42.315D4
 Upper central C1 — C2×C4 — C2×C42 — C42.315D4
 Jennings C1 — C22 — C22 — C42 — C42.315D4

Generators and relations for C42.315D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 292 in 138 conjugacy classes, 56 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C4 [×4], C22, C22 [×2], C22 [×8], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×8], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C4×C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, D4⋊C8 [×4], C2×C4×C8, C4⋊M4(2), C22.26C24, C42.315D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C8○D4 [×2], C2×D8, C2×SD16, (C22×C8)⋊C2, C2×D4⋊C4, C2×C4≀C2, C42.315D4

Smallest permutation representation of C42.315D4
On 64 points
Generators in S64
```(1 9 17 50)(2 10 18 51)(3 11 19 52)(4 12 20 53)(5 13 21 54)(6 14 22 55)(7 15 23 56)(8 16 24 49)(25 48 60 40)(26 41 61 33)(27 42 62 34)(28 43 63 35)(29 44 64 36)(30 45 57 37)(31 46 58 38)(32 47 59 39)
(1 30 21 61)(2 31 22 62)(3 32 23 63)(4 25 24 64)(5 26 17 57)(6 27 18 58)(7 28 19 59)(8 29 20 60)(9 45 54 33)(10 46 55 34)(11 47 56 35)(12 48 49 36)(13 41 50 37)(14 42 51 38)(15 43 52 39)(16 44 53 40)
(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 29 30 20 21 60 61 8)(2 19 31 59 22 7 62 28)(3 58 32 6 23 27 63 18)(4 5 25 26 24 17 64 57)(9 48 45 49 54 36 33 12)(10 56 46 35 55 11 34 47)(13 44 41 53 50 40 37 16)(14 52 42 39 51 15 38 43)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D8 SD16 C4≀C2 C8○D4 kernel C42.315D4 D4⋊C8 C2×C4×C8 C4⋊M4(2) C22.26C24 C4⋊D4 C4⋊1D4 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C4 C4 # reps 1 4 1 1 1 4 2 2 2 2 4 4 8 8

Matrix representation of C42.315D4 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 16 15 0 0 1 1
,
 4 0 0 0 0 4 0 0 0 0 16 0 0 0 0 16
,
 12 12 0 0 11 0 0 0 0 0 0 7 0 0 5 10
,
 12 12 0 0 11 5 0 0 0 0 0 7 0 0 12 0
`G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,1,0,0,15,1],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[12,11,0,0,12,0,0,0,0,0,0,5,0,0,7,10],[12,11,0,0,12,5,0,0,0,0,0,12,0,0,7,0] >;`

C42.315D4 in GAP, Magma, Sage, TeX

`C_4^2._{315}D_4`
`% in TeX`

`G:=Group("C4^2.315D4");`
`// GroupNames label`

`G:=SmallGroup(128,224);`
`// by ID`

`G=gap.SmallGroup(128,224);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,1123,570,136,172]);`
`// Polycyclic`

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

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