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## G = C42.42Q8order 128 = 27

### 2nd non-split extension by C42 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C42.42Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C42.42Q8
 Lower central C1 — C2 — C4 — C42.42Q8
 Upper central C1 — C42 — C2×C42 — C42.42Q8
 Jennings C1 — C22 — C22 — C42 — C42.42Q8

Generators and relations for C42.42Q8
G = < a,b,c,d | a4=b4=1, c4=a2, d2=a2bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 124 in 90 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], C2×C8 [×6], C22×C4 [×3], C4×C8 [×4], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C82C8 [×2], C81C8 [×2], C2×C4×C8, C42.12C4 [×2], C42.42Q8
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], C8.C4 [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C4○D8 [×2], C2×C4⋊C8, C23.25D4, C2×C8.C4, C42.42Q8

Smallest permutation representation of C42.42Q8
On 64 points
Generators in S64
```(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)
(1 33 46 53)(2 34 47 54)(3 35 48 55)(4 36 41 56)(5 37 42 49)(6 38 43 50)(7 39 44 51)(8 40 45 52)(9 64 21 29)(10 57 22 30)(11 58 23 31)(12 59 24 32)(13 60 17 25)(14 61 18 26)(15 62 19 27)(16 63 20 28)
(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 32 39 10 42 63 55 18)(2 27 40 13 43 58 56 21)(3 30 33 16 44 61 49 24)(4 25 34 11 45 64 50 19)(5 28 35 14 46 59 51 22)(6 31 36 9 47 62 52 17)(7 26 37 12 48 57 53 20)(8 29 38 15 41 60 54 23)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 8A ··· 8P 8Q ··· 8AF order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C2 C4 C4 C8 D4 Q8 D4 Q8 M4(2) C8.C4 C4○D8 kernel C42.42Q8 C8⋊2C8 C8⋊1C8 C2×C4×C8 C42.12C4 C4×C8 C22×C8 C2×C8 C42 C42 C22×C4 C22×C4 C2×C4 C4 C4 # reps 1 2 2 1 2 4 4 16 1 1 1 1 4 8 8

Matrix representation of C42.42Q8 in GL3(𝔽17) generated by

 16 0 0 0 13 0 0 0 13
,
 4 0 0 0 4 0 0 0 13
,
 1 0 0 0 15 0 0 0 9
,
 15 0 0 0 0 1 0 1 0
`G:=sub<GL(3,GF(17))| [16,0,0,0,13,0,0,0,13],[4,0,0,0,4,0,0,0,13],[1,0,0,0,15,0,0,0,9],[15,0,0,0,0,1,0,1,0] >;`

C42.42Q8 in GAP, Magma, Sage, TeX

`C_4^2._{42}Q_8`
`% in TeX`

`G:=Group("C4^2.42Q8");`
`// GroupNames label`

`G:=SmallGroup(128,296);`
`// by ID`

`G=gap.SmallGroup(128,296);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,184,1123,136,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a^2*b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;`
`// generators/relations`

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