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## G = C42.293C23order 128 = 27

### 154th 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 — C42.293C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C8⋊C4 — C2×C8⋊C4 — C42.293C23
 Lower central C1 — C22 — C42.293C23
 Upper central C1 — C2×C4 — C42.293C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.293C23

Generators and relations for C42.293C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=a2c, de=ed >

Subgroups: 252 in 186 conjugacy classes, 130 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×5], Q8, C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×4], C8⋊C4 [×2], C8⋊C4 [×4], 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 [×4], C2×M4(2) [×2], C2×C8⋊C4, C82M4(2) [×2], C42.12C4, C42.7C22 [×2], C8×D4 [×2], C89D4 [×4], C84Q8 [×2], C23.36C23, C42.293C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C2×C8○D4, Q8○M4(2), C42.293C23

Smallest permutation representation of C42.293C23
On 64 points
Generators in S64
```(1 42 27 18)(2 19 28 43)(3 44 29 20)(4 21 30 45)(5 46 31 22)(6 23 32 47)(7 48 25 24)(8 17 26 41)(9 55 35 60)(10 61 36 56)(11 49 37 62)(12 63 38 50)(13 51 39 64)(14 57 40 52)(15 53 33 58)(16 59 34 54)
(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 42 19 44 21 46 23 48)(18 43 20 45 22 47 24 41)(25 30 27 32 29 26 31 28)(33 34 35 36 37 38 39 40)(49 59 51 61 53 63 55 57)(50 60 52 62 54 64 56 58)
(1 59)(2 55)(3 61)(4 49)(5 63)(6 51)(7 57)(8 53)(9 43)(10 20)(11 45)(12 22)(13 47)(14 24)(15 41)(16 18)(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)(2,19,28,43)(3,44,29,20)(4,21,30,45)(5,46,31,22)(6,23,32,47)(7,48,25,24)(8,17,26,41)(9,55,35,60)(10,61,36,56)(11,49,37,62)(12,63,38,50)(13,51,39,64)(14,57,40,52)(15,53,33,58)(16,59,34,54), (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)(17,42,19,44,21,46,23,48)(18,43,20,45,22,47,24,41)(25,30,27,32,29,26,31,28)(33,34,35,36,37,38,39,40)(49,59,51,61,53,63,55,57)(50,60,52,62,54,64,56,58), (1,59)(2,55)(3,61)(4,49)(5,63)(6,51)(7,57)(8,53)(9,43)(10,20)(11,45)(12,22)(13,47)(14,24)(15,41)(16,18)(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)(2,19,28,43)(3,44,29,20)(4,21,30,45)(5,46,31,22)(6,23,32,47)(7,48,25,24)(8,17,26,41)(9,55,35,60)(10,61,36,56)(11,49,37,62)(12,63,38,50)(13,51,39,64)(14,57,40,52)(15,53,33,58)(16,59,34,54), (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)(17,42,19,44,21,46,23,48)(18,43,20,45,22,47,24,41)(25,30,27,32,29,26,31,28)(33,34,35,36,37,38,39,40)(49,59,51,61,53,63,55,57)(50,60,52,62,54,64,56,58), (1,59)(2,55)(3,61)(4,49)(5,63)(6,51)(7,57)(8,53)(9,43)(10,20)(11,45)(12,22)(13,47)(14,24)(15,41)(16,18)(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),(2,19,28,43),(3,44,29,20),(4,21,30,45),(5,46,31,22),(6,23,32,47),(7,48,25,24),(8,17,26,41),(9,55,35,60),(10,61,36,56),(11,49,37,62),(12,63,38,50),(13,51,39,64),(14,57,40,52),(15,53,33,58),(16,59,34,54)], [(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),(17,42,19,44,21,46,23,48),(18,43,20,45,22,47,24,41),(25,30,27,32,29,26,31,28),(33,34,35,36,37,38,39,40),(49,59,51,61,53,63,55,57),(50,60,52,62,54,64,56,58)], [(1,59),(2,55),(3,61),(4,49),(5,63),(6,51),(7,57),(8,53),(9,43),(10,20),(11,45),(12,22),(13,47),(14,24),(15,41),(16,18),(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)])`

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4○D4 C8○D4 Q8○M4(2) kernel C42.293C23 C2×C8⋊C4 C8○2M4(2) C42.12C4 C42.7C22 C8×D4 C8⋊9D4 C8⋊4Q8 C23.36C23 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C8 C22 C2 # reps 1 1 2 1 2 2 4 2 1 2 2 4 2 2 4 8 8 2

Matrix representation of C42.293C23 in GL4(𝔽17) generated by

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

C42.293C23 in GAP, Magma, Sage, TeX

`C_4^2._{293}C_2^3`
`% in TeX`

`G:=Group("C4^2.293C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1700);`
`// by ID`

`G=gap.SmallGroup(128,1700);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,80,124]);`
`// 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,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,d*e=e*d>;`
`// generators/relations`

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