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## G = C42.28C23order 128 = 27

### 28th non-split extension by C42 of C23 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.28C23
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C22.35C24 — C42.28C23
 Lower central C1 — C2 — C2×C4 — C42.28C23
 Upper central C1 — C22 — C42⋊C2 — C42.28C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.28C23

Generators and relations for C42.28C23
G = < a,b,c,d,e | a4=b4=1, c2=d2=e2=b2, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=a2d >

Subgroups: 268 in 161 conjugacy classes, 84 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×8], C23, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×16], C2×C8 [×4], C2×C8, M4(2), Q16 [×4], C22×C4, C2×Q8 [×4], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C42.C2 [×4], C422C2 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C42.6C22, C42Q16 [×4], C8.18D4 [×2], C8.D4 [×2], Q8.Q8 [×4], C22.35C24 [×2], C42.28C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, Q8○D8 [×2], C42.28C23

Character table of C42.28C23

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 2 2 4 4 4 4 4 8 8 8 8 8 8 8 8 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ5 1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ10 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ11 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ12 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ13 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ14 1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ16 1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ17 2 2 2 2 -2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 -2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 4 -4 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2

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

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

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

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

Matrix representation of C42.28C23 in GL8(𝔽17)

 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 1 1
,
 0 0 1 10 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 12 12 0 7 0 0 0 0 5 5 10 10 0 0 0 0 5 5 0 0 0 0 0 0 12 0 12 0
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 16 16 15 0 0 0 0 0 1 1 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 4 4 8 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 13

`G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,0,0,0,0,0,0,12,5,5,12,0,0,0,0,12,5,5,0,0,0,0,0,0,10,0,12,0,0,0,0,7,10,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,16,1,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,13,0,0,0,0,0,4,13,0,0,0,0,0,0,8,0,0,13] >;`

C42.28C23 in GAP, Magma, Sage, TeX

`C_4^2._{28}C_2^3`
`% in TeX`

`G:=Group("C4^2.28C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1993);`
`// by ID`

`G=gap.SmallGroup(128,1993);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,352,675,1018,80,4037,1027,124]);`
`// Polycyclic`

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

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