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## G = C42.8C23order 128 = 27

### 8th 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 — C42 — C42.8C23
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4.4D4 — C22.56C24 — C42.8C23
 Lower central C1 — C22 — C42 — C42.8C23
 Upper central C1 — C22 — C42 — C42.8C23
 Jennings C1 — C22 — C22 — C42 — C42.8C23

Generators and relations for C42.8C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, eae=a-1b2, cbc-1=ebe=b-1, dbd-1=a2b-1, dcd-1=ac, ece=bc, de=ed >

Subgroups: 248 in 98 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22, C22 [×6], C8 [×3], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×4], Q8 [×4], C23 [×2], C42, C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C8⋊C4, C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2, C4⋊Q8, C42.C22 [×2], C42.2C22, C42.28C22 [×2], C42.30C22, C22.56C24, C42.8C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D4.8D4, D4.9D4 [×2], C42.8C23

Character table of C42.8C23

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F size 1 1 1 1 8 8 4 4 4 8 8 8 8 16 8 8 8 8 8 8 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 2 0 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 2 -2 0 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 0 -2 -2 2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i complex lifted from D4.8D4 ρ16 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from D4.9D4 ρ17 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from D4.9D4 ρ18 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from D4.9D4 ρ19 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i complex lifted from D4.8D4 ρ20 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from D4.9D4

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

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

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

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

Matrix representation of C42.8C23 in GL8(𝔽17)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 16 0
,
 2 2 15 2 0 0 0 0 2 2 2 15 0 0 0 0 15 2 15 15 0 0 0 0 2 15 15 15 0 0 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 12 12 0 0 0 0 5 12 0 0 0 0 0 0 12 12 0 0
,
 16 6 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 6 16 0 0 0 0 0 0 0 0 3 0 14 0 0 0 0 0 0 3 0 14 0 0 0 0 14 0 14 0 0 0 0 0 0 14 0 14
,
 1 0 0 0 0 0 0 0 0 1 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 1 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 16

`G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[2,2,15,2,0,0,0,0,2,2,2,15,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0],[16,11,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,11,16,0,0,0,0,0,0,0,0,3,0,14,0,0,0,0,0,0,3,0,14,0,0,0,0,14,0,14,0,0,0,0,0,0,14,0,14],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;`

C42.8C23 in GAP, Magma, Sage, TeX

`C_4^2._8C_2^3`
`% in TeX`

`G:=Group("C4^2.8C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,394);`
`// by ID`

`G=gap.SmallGroup(128,394);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,456,422,1123,570,521,136,3924,1411,998,242]);`
`// Polycyclic`

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

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