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## G = C42.294C23order 128 = 27

### 155th 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 — C42.294C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C8⋊C4 — C4×M4(2) — C42.294C23
 Lower central C1 — C22 — C42.294C23
 Upper central C1 — C2×C4 — C42.294C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.294C23

Generators and relations for C42.294C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b, e2=a2, ab=ba, cac-1=a-1b2, dad-1=eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2c, ce=ec, de=ed >

Subgroups: 252 in 185 conjugacy classes, 130 normal (52 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×2], C4×C8 [×4], C8⋊C4 [×2], C8⋊C4 [×2], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×4], 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) [×2], C2×M4(2) [×2], C4×M4(2), C82M4(2) [×2], C42.12C4, C42.7C22 [×2], C8×D4, C89D4 [×2], C86D4, C86D4 [×2], C8×Q8, C84Q8, C23.36C23, C42.294C23
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), C42.294C23

Smallest permutation representation of C42.294C23
On 64 points
Generators in S64
```(1 37 24 48)(2 34 17 45)(3 39 18 42)(4 36 19 47)(5 33 20 44)(6 38 21 41)(7 35 22 46)(8 40 23 43)(9 55 32 60)(10 52 25 57)(11 49 26 62)(12 54 27 59)(13 51 28 64)(14 56 29 61)(15 53 30 58)(16 50 31 63)
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 64 13 60)(10 52 14 56)(11 58 15 62)(12 54 16 50)(17 41 21 45)(18 39 22 35)(19 43 23 47)(20 33 24 37)(25 57 29 61)(26 53 30 49)(27 59 31 63)(28 55 32 51)
(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 59 24 54)(2 60 17 55)(3 61 18 56)(4 62 19 49)(5 63 20 50)(6 64 21 51)(7 57 22 52)(8 58 23 53)(9 41 32 38)(10 42 25 39)(11 43 26 40)(12 44 27 33)(13 45 28 34)(14 46 29 35)(15 47 30 36)(16 48 31 37)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4L 4M ··· 4S 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 2 2 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 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 1 1 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4○D4 C8○D4 Q8○M4(2) kernel C42.294C23 C4×M4(2) C8○2M4(2) C42.12C4 C42.7C22 C8×D4 C8⋊9D4 C8⋊6D4 C8×Q8 C8⋊4Q8 C23.36C23 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C8 C4 C2 # reps 1 1 2 1 2 1 2 3 1 1 1 2 2 4 2 2 4 8 8 2

Matrix representation of C42.294C23 in GL4(𝔽17) generated by

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

C42.294C23 in GAP, Magma, Sage, TeX

`C_4^2._{294}C_2^3`
`% in TeX`

`G:=Group("C4^2.294C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1701);`
`// by ID`

`G=gap.SmallGroup(128,1701);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,521,80,124]);`
`// Polycyclic`

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

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