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## G = C4⋊C4.7C8order 128 = 27

### 4th non-split extension by C4⋊C4 of C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4⋊C4.7C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C8○2M4(2) — C4⋊C4.7C8
 Lower central C1 — C22 — C4⋊C4.7C8
 Upper central C1 — C2×C8 — C4⋊C4.7C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C4⋊C4.7C8

Generators and relations for C4⋊C4.7C8
G = < a,b,c | a4=b4=1, c8=a2, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 100 in 80 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], C23, C16 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×4], C2×C16 [×2], M5(2) [×2], C42⋊C2, C22×C8, C2×M4(2), C4⋊C16 [×4], C82M4(2), C22×C16, C2×M5(2), C4⋊C4.7C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, D4○C16 [×2], C4⋊C4.7C8

Smallest permutation representation of C4⋊C4.7C8
On 64 points
Generators in S64
```(1 46 9 38)(2 47 10 39)(3 48 11 40)(4 33 12 41)(5 34 13 42)(6 35 14 43)(7 36 15 44)(8 37 16 45)(17 53 25 61)(18 54 26 62)(19 55 27 63)(20 56 28 64)(21 57 29 49)(22 58 30 50)(23 59 31 51)(24 60 32 52)
(1 19 42 59)(2 52 43 28)(3 21 44 61)(4 54 45 30)(5 23 46 63)(6 56 47 32)(7 25 48 49)(8 58 33 18)(9 27 34 51)(10 60 35 20)(11 29 36 53)(12 62 37 22)(13 31 38 55)(14 64 39 24)(15 17 40 57)(16 50 41 26)
(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)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8H 8I 8J 8K 8L 8M 8N 8O 8P 16A ··· 16P 16Q ··· 16X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 8 8 8 8 16 ··· 16 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 D4 Q8 M4(2) D4○C16 kernel C4⋊C4.7C8 C4⋊C16 C8○2M4(2) C22×C16 C2×M5(2) C8⋊C4 C42⋊C2 C2×M4(2) C22⋊C4 C4⋊C4 C2×C8 C2×C8 C2×C4 C2 # reps 1 4 1 1 1 4 2 2 8 8 2 2 4 16

Matrix representation of C4⋊C4.7C8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 4 0 0 0 16 13
,
 16 16 0 0 2 1 0 0 0 0 1 8 0 0 4 16
,
 16 16 0 0 0 1 0 0 0 0 12 0 0 0 0 12
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,16,0,0,0,13],[16,2,0,0,16,1,0,0,0,0,1,4,0,0,8,16],[16,0,0,0,16,1,0,0,0,0,12,0,0,0,0,12] >;`

C4⋊C4.7C8 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._7C_8`
`% in TeX`

`G:=Group("C4:C4.7C8");`
`// GroupNames label`

`G:=SmallGroup(128,883);`
`// by ID`

`G=gap.SmallGroup(128,883);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=1,c^8=a^2,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;`
`// generators/relations`

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