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## G = C42.110D4order 128 = 27

### 92nd 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 — C42.110D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.110D4
 Lower central C1 — C2 — C2×C4 — C42.110D4
 Upper central C1 — C23 — C2×C42 — C42.110D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.110D4

Generators and relations for C42.110D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 388 in 172 conjugacy classes, 64 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×C8 [×2], C22×D4, C22×Q8, C429C4, C2×C8⋊C4, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C2×C4.4D4, C42.110D4
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, C8⋊C22 [×2], C8.C22 [×2], C24.3C22, C23.36D4 [×2], C42.28C22 [×2], C83D4, C8.2D4, C42.110D4

Smallest permutation representation of C42.110D4
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 38 29 13)(2 39 30 14)(3 40 31 15)(4 37 32 16)(5 19 59 44)(6 20 60 41)(7 17 57 42)(8 18 58 43)(9 49 34 25)(10 50 35 26)(11 51 36 27)(12 52 33 28)(21 62 46 53)(22 63 47 54)(23 64 48 55)(24 61 45 56)
(1 55 9 44)(2 54 10 43)(3 53 11 42)(4 56 12 41)(5 13 23 25)(6 16 24 28)(7 15 21 27)(8 14 22 26)(17 31 62 36)(18 30 63 35)(19 29 64 34)(20 32 61 33)(37 45 52 60)(38 48 49 59)(39 47 50 58)(40 46 51 57)
(1 4 29 32)(2 31 30 3)(5 61 59 56)(6 55 60 64)(7 63 57 54)(8 53 58 62)(9 12 34 33)(10 36 35 11)(13 37 38 16)(14 15 39 40)(17 22 42 47)(18 46 43 21)(19 24 44 45)(20 48 41 23)(25 52 49 28)(26 27 50 51)```

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

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

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

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 1 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C8⋊C22 C8.C22 kernel C42.110D4 C42⋊9C4 C2×C8⋊C4 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.4D4 C4.4D4 C42 C2×C8 C22×C4 C2×C4 C22 C22 # reps 1 1 1 2 2 1 8 2 4 2 4 2 2

Matrix representation of C42.110D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 16
,
 1 13 0 0 0 0 0 0 9 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 13 13 0 0 0 0 4 0 0 0 0 0 0 0 13 13 0 0
,
 16 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 13 9 0 0 0 0 0 0 0 4

`G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,9,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0],[16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4] >;`

C42.110D4 in GAP, Magma, Sage, TeX

`C_4^2._{110}D_4`
`% in TeX`

`G:=Group("C4^2.110D4");`
`// GroupNames label`

`G:=SmallGroup(128,691);`
`// by ID`

`G=gap.SmallGroup(128,691);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,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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