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G = C42.694C23order 128 = 27

109th 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 — C42.694C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.26C24 — C42.694C23
 Lower central C1 — C22 — C42.694C23
 Upper central C1 — C2×C4 — C42.694C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.694C23

Generators and relations for C42.694C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1b2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 332 in 205 conjugacy classes, 128 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×15], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.12C4 [×2], C8×D4 [×4], C86D4 [×4], C22.26C24, C42.694C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×4], C23×C4, 2+ 1+4 [×2], C22.11C24, C2×C8○D4 [×2], C42.694C23

Smallest permutation representation of C42.694C23
On 64 points
Generators in S64
```(1 10 55 57)(2 11 56 58)(3 12 49 59)(4 13 50 60)(5 14 51 61)(6 15 52 62)(7 16 53 63)(8 9 54 64)(17 45 29 39)(18 46 30 40)(19 47 31 33)(20 48 32 34)(21 41 25 35)(22 42 26 36)(23 43 27 37)(24 44 28 38)
(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)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)
(1 33 55 47)(2 48 56 34)(3 35 49 41)(4 42 50 36)(5 37 51 43)(6 44 52 38)(7 39 53 45)(8 46 54 40)(9 30 64 18)(10 19 57 31)(11 32 58 20)(12 21 59 25)(13 26 60 22)(14 23 61 27)(15 28 62 24)(16 17 63 29)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2H 4A 4B 4C 4D 4E ··· 4L 4M ··· 4Q 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 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 2 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8○D4 2+ 1+4 kernel C42.694C23 (C22×C8)⋊C2 C42.12C4 C8×D4 C8⋊6D4 C22.26C24 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C4 C4 # reps 1 4 2 4 4 1 8 4 2 2 16 2

Matrix representation of C42.694C23 in GL4(𝔽17) generated by

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

C42.694C23 in GAP, Magma, Sage, TeX

`C_4^2._{694}C_2^3`
`% in TeX`

`G:=Group("C4^2.694C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1711);`
`// by ID`

`G=gap.SmallGroup(128,1711);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,1018,521,124]);`
`// Polycyclic`

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

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