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## G = C42.698C23order 128 = 27

### 113rd 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 — C42.698C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C4×C4○D4 — C42.698C23
 Lower central C1 — C22 — C42.698C23
 Upper central C1 — C2×C4 — C42.698C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.698C23

Generators and relations for C42.698C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ece=b2c, ede=a2d >

Subgroups: 276 in 194 conjugacy classes, 134 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×10], C2×C4 [×13], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×C4○D4, C2×C4⋊C8, C4⋊M4(2) [×2], C42.12C4, C42.6C4 [×2], C89D4 [×4], C86D4 [×2], C84Q8 [×2], C4×C4○D4, C42.698C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C22×M4(2), Q8○M4(2), C42.698C23

Smallest permutation representation of C42.698C23
On 64 points
Generators in S64
```(1 31 55 12)(2 13 56 32)(3 25 49 14)(4 15 50 26)(5 27 51 16)(6 9 52 28)(7 29 53 10)(8 11 54 30)(17 60 40 46)(18 47 33 61)(19 62 34 48)(20 41 35 63)(21 64 36 42)(22 43 37 57)(23 58 38 44)(24 45 39 59)
(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 41)(2 64)(3 43)(4 58)(5 45)(6 60)(7 47)(8 62)(9 40)(10 18)(11 34)(12 20)(13 36)(14 22)(15 38)(16 24)(17 28)(19 30)(21 32)(23 26)(25 37)(27 39)(29 33)(31 35)(42 56)(44 50)(46 52)(48 54)(49 57)(51 59)(53 61)(55 63)
(2 6)(4 8)(9 13)(11 15)(17 36)(18 33)(19 38)(20 35)(21 40)(22 37)(23 34)(24 39)(26 30)(28 32)(41 63)(42 60)(43 57)(44 62)(45 59)(46 64)(47 61)(48 58)(50 54)(52 56)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 8A ··· 8P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 M4(2) 2+ 1+4 2- 1+4 Q8○M4(2) kernel C42.698C23 C2×C4⋊C8 C4⋊M4(2) C42.12C4 C42.6C4 C8⋊9D4 C8⋊6D4 C8⋊4Q8 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 4 2 2 1 6 6 2 2 8 1 1 2

Matrix representation of C42.698C23 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 14 15 0 0 0 0 15 3 0 0 0 0 0 0 0 0 2 14 0 0 0 0 14 15 0 0 15 3 0 0 0 0 3 2 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 14 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[14,15,0,0,0,0,15,3,0,0,0,0,0,0,0,0,15,3,0,0,0,0,3,2,0,0,2,14,0,0,0,0,14,15,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,14,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C42.698C23 in GAP, Magma, Sage, TeX

`C_4^2._{698}C_2^3`
`% in TeX`

`G:=Group("C4^2.698C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1721);`
`// by ID`

`G=gap.SmallGroup(128,1721);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,80,124]);`
`// Polycyclic`

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

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