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## G = C42.696C23order 128 = 27

### 111st non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.696C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C42.696C23
 Lower central C1 — C22 — C42.696C23
 Upper central C1 — C2×C4 — C42.696C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.696C23

Generators and relations for C42.696C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=a2c, ede=b2d >

Subgroups: 204 in 161 conjugacy classes, 126 normal (28 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×12], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×8], C2×C4 [×5], Q8 [×6], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C22×C4 [×3], C2×Q8 [×4], C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×6], C22⋊C8 [×4], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C42.12C4, C42.6C4, C42.7C22 [×4], C8×Q8 [×2], C84Q8 [×6], C23.37C23, C42.696C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, 2- 1+4 [×2], C23.32C23, C2×C8○D4, Q8○M4(2), C42.696C23

Smallest permutation representation of C42.696C23
On 64 points
Generators in S64
```(1 31 55 23)(2 32 56 24)(3 25 49 17)(4 26 50 18)(5 27 51 19)(6 28 52 20)(7 29 53 21)(8 30 54 22)(9 58 38 42)(10 59 39 43)(11 60 40 44)(12 61 33 45)(13 62 34 46)(14 63 35 47)(15 64 36 48)(16 57 37 41)
(1 3 5 7)(2 4 6 8)(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 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 55 47)(2 48 56 64)(3 57 49 41)(4 42 50 58)(5 59 51 43)(6 44 52 60)(7 61 53 45)(8 46 54 62)(9 26 38 18)(10 19 39 27)(11 28 40 20)(12 21 33 29)(13 30 34 22)(14 23 35 31)(15 32 36 24)(16 17 37 25)
(2 56)(4 50)(6 52)(8 54)(9 34)(10 14)(11 36)(12 16)(13 38)(15 40)(18 26)(20 28)(22 30)(24 32)(33 37)(35 39)(41 45)(42 62)(43 47)(44 64)(46 58)(48 60)(57 61)(59 63)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C8○D4 2- 1+4 Q8○M4(2) kernel C42.696C23 C42.12C4 C42.6C4 C42.7C22 C8×Q8 C8⋊4Q8 C23.37C23 C22⋊Q8 C42.C2 C4⋊Q8 C4 C4 C2 # reps 1 1 1 4 2 6 1 8 4 4 8 2 2

Matrix representation of C42.696C23 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 0 0 0 0 2 1 0 0 0 0 9 7 0 1 0 0 12 2 16 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 10 14 15 0 0 0 7 13 2 2 0 0 5 8 4 14 0 0 8 12 7 7
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 10 0 0 0 0 0 11 15 5 5 0 0 10 8 5 12
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 10 14 16 0 0 0 0 7 0 16

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,2,9,12,0,0,16,1,7,2,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,5,8,0,0,14,13,8,12,0,0,15,2,4,7,0,0,0,2,14,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,10,11,10,0,0,5,0,15,8,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,10,0,0,0,0,1,14,7,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C42.696C23 in GAP, Magma, Sage, TeX

`C_4^2._{696}C_2^3`
`% in TeX`

`G:=Group("C4^2.696C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1717);`
`// by ID`

`G=gap.SmallGroup(128,1717);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,219,100,675,1018,124]);`
`// Polycyclic`

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

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