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## G = C42.292C23order 128 = 27

### 153rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.292C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C4×M4(2) — C42.292C23
 Lower central C1 — C22 — C42.292C23
 Upper central C1 — C2×C4 — C42.292C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.292C23

Generators and relations for C42.292C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, ede=b2d >

Subgroups: 252 in 181 conjugacy classes, 128 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×3], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×2], C8⋊C4 [×6], C22⋊C8 [×2], C22⋊C8 [×4], C4⋊C8 [×6], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C4×M4(2), C82M4(2) [×2], C42.6C4, C42.7C22 [×2], C89D4 [×4], C86D4 [×2], C84Q8 [×2], C23.36C23, C42.292C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], C4×C4○D4, Q8○M4(2) [×2], C42.292C23

Smallest permutation representation of C42.292C23
On 64 points
Generators in S64
```(1 42 27 18)(2 23 28 47)(3 44 29 20)(4 17 30 41)(5 46 31 22)(6 19 32 43)(7 48 25 24)(8 21 26 45)(9 53 37 58)(10 63 38 50)(11 55 39 60)(12 57 40 52)(13 49 33 62)(14 59 34 54)(15 51 35 64)(16 61 36 56)
(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 6 3 8 5 2 7 4)(9 10 11 12 13 14 15 16)(17 46 19 48 21 42 23 44)(18 47 20 41 22 43 24 45)(25 30 27 32 29 26 31 28)(33 34 35 36 37 38 39 40)(49 63 51 57 53 59 55 61)(50 64 52 58 54 60 56 62)
(1 59)(2 55)(3 61)(4 49)(5 63)(6 51)(7 57)(8 53)(9 45)(10 22)(11 47)(12 24)(13 41)(14 18)(15 43)(16 20)(17 33)(19 35)(21 37)(23 39)(25 52)(26 58)(27 54)(28 60)(29 56)(30 62)(31 50)(32 64)(34 42)(36 44)(38 46)(40 48)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4Q 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 4 4 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 1 1 1 1 1 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4○D4 Q8○M4(2) kernel C42.292C23 C4×M4(2) C8○2M4(2) C42.6C4 C42.7C22 C8⋊9D4 C8⋊6D4 C8⋊4Q8 C23.36C23 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C8 C2 # reps 1 1 2 1 2 4 2 2 1 2 2 4 2 2 4 8 4

Matrix representation of C42.292C23 in GL6(𝔽17)

 15 9 0 0 0 0 7 2 0 0 0 0 0 0 4 8 3 1 0 0 0 12 0 1 0 0 12 5 13 0 0 0 0 10 0 5
,
 1 0 0 0 0 0 0 1 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
,
 16 0 0 0 0 0 9 1 0 0 0 0 0 0 0 3 16 10 0 0 0 3 0 9 0 0 4 4 0 14 0 0 0 8 0 14
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 3 1 9 0 0 0 3 0 9 0 0 13 4 0 0 0 0 0 8 0 14
,
 9 2 0 0 0 0 11 8 0 0 0 0 0 0 8 4 3 1 0 0 12 0 6 1 0 0 5 12 4 0 0 0 10 0 8 5

`G:=sub<GL(6,GF(17))| [15,7,0,0,0,0,9,2,0,0,0,0,0,0,4,0,12,0,0,0,8,12,5,10,0,0,3,0,13,0,0,0,1,1,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,3,3,4,8,0,0,16,0,0,0,0,0,10,9,14,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,3,3,4,8,0,0,1,0,0,0,0,0,9,9,0,14],[9,11,0,0,0,0,2,8,0,0,0,0,0,0,8,12,5,10,0,0,4,0,12,0,0,0,3,6,4,8,0,0,1,1,0,5] >;`

C42.292C23 in GAP, Magma, Sage, TeX

`C_4^2._{292}C_2^3`
`% in TeX`

`G:=Group("C4^2.292C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1699);`
`// by ID`

`G=gap.SmallGroup(128,1699);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,80,124]);`
`// Polycyclic`

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

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