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## G = C42.119D4order 128 = 27

### 101st 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 — C42.119D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.119D4
 Lower central C1 — C2 — C2×C4 — C42.119D4
 Upper central C1 — C23 — C2×C42 — C42.119D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.119D4

Generators and relations for C42.119D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 380 in 172 conjugacy classes, 64 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×10], C22 [×3], C22 [×4], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], 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], C4⋊C8 [×2], C2×C42, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×C8 [×2], C22×D4, C22×Q8, C4×C4⋊C4, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C2×C4⋊C8, C2×C4.4D4, C42.119D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C4○D8 [×2], C8⋊C22, C8.C22, C24.3C22, C23.24D4, C23.36D4, D4.2D4 [×2], Q8.D4 [×2], C42.119D4

Smallest permutation representation of C42.119D4
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 29 25 50)(2 30 26 51)(3 31 27 52)(4 32 28 49)(5 53 61 60)(6 54 62 57)(7 55 63 58)(8 56 64 59)(9 39 13 36)(10 40 14 33)(11 37 15 34)(12 38 16 35)(17 48 24 44)(18 45 21 41)(19 46 22 42)(20 47 23 43)
(1 21 39 60)(2 22 40 57)(3 23 37 58)(4 24 38 59)(5 50 41 9)(6 51 42 10)(7 52 43 11)(8 49 44 12)(13 61 29 45)(14 62 30 46)(15 63 31 47)(16 64 32 48)(17 35 56 28)(18 36 53 25)(19 33 54 26)(20 34 55 27)
(1 12 25 16)(2 15 26 11)(3 10 27 14)(4 13 28 9)(5 64 61 8)(6 7 62 63)(17 21 24 18)(19 23 22 20)(29 35 50 38)(30 37 51 34)(31 33 52 40)(32 39 49 36)(41 48 45 44)(42 43 46 47)(53 56 60 59)(54 58 57 55)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 4S 4T 8A ··· 8H order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 4 ··· 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C42.119D4 C4×C4⋊C4 C2×D4⋊C4 C2×Q8⋊C4 C2×C4⋊C8 C2×C4.4D4 C4.4D4 C42 C4⋊C4 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 2 2 1 1 8 2 4 2 4 8 1 1

Matrix representation of C42.119D4 in GL5(𝔽17)

 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 16 0
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 5 12 0 0 0 12 12 0 0 0 0 0 0 4 0 0 0 13 0
,
 16 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 0 16 0 0 0 16 0

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

C42.119D4 in GAP, Magma, Sage, TeX

`C_4^2._{119}D_4`
`% in TeX`

`G:=Group("C4^2.119D4");`
`// GroupNames label`

`G:=SmallGroup(128,715);`
`// by ID`

`G=gap.SmallGroup(128,715);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,1018,248,2028,124]);`
`// Polycyclic`

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

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