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## G = C42.291C23order 128 = 27

### 152nd 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 — C42.291C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C2×C4×C8 — C42.291C23
 Lower central C1 — C22 — C42.291C23
 Upper central C1 — C2×C8 — C42.291C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.291C23

Generators and relations for C42.291C23
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede-1=b2d >

Subgroups: 252 in 190 conjugacy classes, 132 normal (52 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×6], C2×C4 [×10], D4 [×6], 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 [×8], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4×C8 [×4], C4×C8 [×2], C8⋊C4 [×4], 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], C22×C8 [×2], C2×M4(2) [×2], C2×C4×C8, C82M4(2) [×2], C42.6C4, C42.7C22 [×2], C8×D4, C8×D4 [×2], C89D4 [×2], C86D4, C8×Q8, C84Q8, C23.36C23, C42.291C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C8○D4 [×4], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C2×C8○D4 [×2], C42.291C23

Smallest permutation representation of C42.291C23
On 64 points
Generators in S64
```(1 33 57 48)(2 34 58 41)(3 35 59 42)(4 36 60 43)(5 37 61 44)(6 38 62 45)(7 39 63 46)(8 40 64 47)(9 51 28 17)(10 52 29 18)(11 53 30 19)(12 54 31 20)(13 55 32 21)(14 56 25 22)(15 49 26 23)(16 50 27 24)
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(17 49 21 53)(18 50 22 54)(19 51 23 55)(20 52 24 56)(33 46 37 42)(34 47 38 43)(35 48 39 44)(36 41 40 45)
(1 48 5 44)(2 38 6 34)(3 42 7 46)(4 40 8 36)(9 21 13 17)(10 52 14 56)(11 23 15 19)(12 54 16 50)(18 25 22 29)(20 27 24 31)(26 53 30 49)(28 55 32 51)(33 61 37 57)(35 63 39 59)(41 58 45 62)(43 60 47 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 24 61 54)(2 21 62 51)(3 18 63 56)(4 23 64 53)(5 20 57 50)(6 17 58 55)(7 22 59 52)(8 19 60 49)(9 45 32 34)(10 42 25 39)(11 47 26 36)(12 44 27 33)(13 41 28 38)(14 46 29 35)(15 43 30 40)(16 48 31 37)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 8A ··· 8H 8I ··· 8T 8U ··· 8AB order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4○D4 C8○D4 C8○D4 kernel C42.291C23 C2×C4×C8 C8○2M4(2) C42.6C4 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 C22 # reps 1 1 2 1 2 3 2 1 1 1 1 2 2 4 2 2 4 8 8 8

Matrix representation of C42.291C23 in GL4(𝔽17) generated by

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

C42.291C23 in GAP, Magma, Sage, TeX

`C_4^2._{291}C_2^3`
`% in TeX`

`G:=Group("C4^2.291C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1698);`
`// by ID`

`G=gap.SmallGroup(128,1698);`
`# by ID`

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

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

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