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## G = C4⋊C4.94D4order 128 = 27

### 49th non-split extension by C4⋊C4 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 — C22×C4 — C4⋊C4.94D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C2×D4⋊C4 — C4⋊C4.94D4
 Lower central C1 — C2 — C22×C4 — C4⋊C4.94D4
 Upper central C1 — C23 — C2×C42 — C4⋊C4.94D4
 Jennings C1 — C2 — C2 — C22×C4 — C4⋊C4.94D4

Generators and relations for C4⋊C4.94D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b-1, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 376 in 159 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×2], C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42, D4⋊C4 [×4], Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4.4D4 [×4], C22×C8 [×2], C22×D4, C22×Q8, C22.7C42, C23.65C23, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C2×C4.4D4, C4⋊C4.94D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C4○D8 [×2], C8⋊C22, C8.C22, C23.10D4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, C42.78C22, C42.28C22, C4⋊C4.94D4

Smallest permutation representation of C4⋊C4.94D4
On 64 points
Generators in S64
```(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 63 11 58)(2 62 12 57)(3 61 9 60)(4 64 10 59)(5 15 21 52)(6 14 22 51)(7 13 23 50)(8 16 24 49)(17 34 39 53)(18 33 40 56)(19 36 37 55)(20 35 38 54)(25 44 30 45)(26 43 31 48)(27 42 32 47)(28 41 29 46)
(1 33 7 44)(2 34 8 41)(3 35 5 42)(4 36 6 43)(9 54 21 47)(10 55 22 48)(11 56 23 45)(12 53 24 46)(13 27 63 20)(14 28 64 17)(15 25 61 18)(16 26 62 19)(29 59 39 51)(30 60 40 52)(31 57 37 49)(32 58 38 50)
(1 42 3 44)(2 41 4 43)(5 33 7 35)(6 36 8 34)(9 45 11 47)(10 48 12 46)(13 39 15 37)(14 38 16 40)(17 52 19 50)(18 51 20 49)(21 56 23 54)(22 55 24 53)(25 59 27 57)(26 58 28 60)(29 61 31 63)(30 64 32 62)```

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C4⋊C4.94D4 C22.7C42 C23.65C23 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.4D4 C4⋊C4 C22×C4 C2×D4 C2×Q8 C2×C4 C22 C22 C22 # reps 1 1 1 2 2 1 2 2 2 2 6 8 1 1

Matrix representation of C4⋊C4.94D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 13 0 0 0 0 0 16 4 0 0 0 0 0 0 3 3 0 0 0 0 3 14
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 1 9 0 0 0 0 13 16 0 0 0 0 0 0 0 13 0 0 0 0 4 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 9 0 0 0 0 0 16 0 0 0 0 0 0 0 13 0 0 0 0 13 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,16,0,0,0,0,0,4,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,13,0,0,0,0,9,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,9,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;`

C4⋊C4.94D4 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{94}D_4`
`% in TeX`

`G:=Group("C4:C4.94D4");`
`// GroupNames label`

`G:=SmallGroup(128,774);`
`// by ID`

`G=gap.SmallGroup(128,774);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2804,1411,718,172,4037,2028,1027,124]);`
`// Polycyclic`

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

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