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## G = C42.52D4order 128 = 27

### 34th 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 — C42.52D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C23.36C23 — C42.52D4
 Lower central C1 — C2 — C2×C4 — C42.52D4
 Upper central C1 — C2×C4 — C2×C42 — C42.52D4
 Jennings C1 — C22 — C22 — C42 — C42.52D4

Generators and relations for C42.52D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 212 in 108 conjugacy classes, 46 normal (36 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×M4(2), D4⋊C8 [×2], Q8⋊C8 [×2], C4⋊M4(2), C42.12C4, C23.36C23, C42.52D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], C8⋊C22, C8.C22, (C22×C8)⋊C2, C23.36D4, C42⋊C22, C42.52D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 C8○D4 C8⋊C22 C8.C22 C42⋊C22 kernel C42.52D4 D4⋊C8 Q8⋊C8 C4⋊M4(2) C42.12C4 C23.36C23 C4⋊D4 C22⋊Q8 C4.4D4 C42.C2 C42 C22×C4 C4 C4 C4 C2 # reps 1 2 2 1 1 1 2 2 2 2 2 2 8 1 1 2

Matrix representation of C42.52D4 in GL6(𝔽17)

 16 15 0 0 0 0 0 1 0 0 0 0 0 0 0 4 16 15 0 0 0 4 15 15 0 0 16 1 0 0 0 0 1 16 13 13
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 5 15 4 4 0 0 5 0 4 0 0 0 13 12 12 12 0 0 12 0 2 0
,
 8 0 0 0 0 0 9 9 0 0 0 0 0 0 5 15 4 4 0 0 5 13 4 8 0 0 13 12 12 12 0 0 12 14 2 4

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,4,4,1,16,0,0,16,15,0,13,0,0,15,15,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,5,13,12,0,0,15,0,12,0,0,0,4,4,12,2,0,0,4,0,12,0],[8,9,0,0,0,0,0,9,0,0,0,0,0,0,5,5,13,12,0,0,15,13,12,14,0,0,4,4,12,2,0,0,4,8,12,4] >;`

C42.52D4 in GAP, Magma, Sage, TeX

`C_4^2._{52}D_4`
`% in TeX`

`G:=Group("C4^2.52D4");`
`// GroupNames label`

`G:=SmallGroup(128,227);`
`// by ID`

`G=gap.SmallGroup(128,227);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1059,1123,570,136,172]);`
`// 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,c*a*c^-1=a^-1,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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