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## G = C4.192+ 1+4order 128 = 27

### 19th non-split extension by C4 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.192+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24 — C4.192+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.192+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.192+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.192+ 1+4

Generators and relations for C4.192+ 1+4
G = < a,b,c,d,e | a4=1, b4=c2=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >

Subgroups: 444 in 201 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C4.Q8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C2×C4○D4, (C22×C8)⋊C2, D4⋊D4, D4.7D4, C88D4, C82D4, C8.D4, C23.46D4, C23.47D4, C22.31C24, C4.192+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○SD16, C4.192+ 1+4

Character table of C4.192+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 8 8 2 2 4 4 4 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 0 0 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 -2 2 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 2 2 0 0 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 -2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 0 0 4 -4 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 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 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 complex lifted from D4○SD16 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 complex lifted from D4○SD16 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 complex lifted from D4○SD16 ρ26 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 complex lifted from D4○SD16

Smallest permutation representation of C4.192+ 1+4
On 64 points
Generators in S64
```(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 25 13 29)(10 26 14 30)(11 27 15 31)(12 28 16 32)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(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 51 5 55)(2 19 6 23)(3 53 7 49)(4 21 8 17)(9 34 13 38)(10 43 14 47)(11 36 15 40)(12 45 16 41)(18 57 22 61)(20 59 24 63)(25 44 29 48)(26 37 30 33)(27 46 31 42)(28 39 32 35)(50 64 54 60)(52 58 56 62)
(1 47 61 39)(2 34 62 42)(3 45 63 37)(4 40 64 48)(5 43 57 35)(6 38 58 46)(7 41 59 33)(8 36 60 44)(9 52 31 19)(10 22 32 55)(11 50 25 17)(12 20 26 53)(13 56 27 23)(14 18 28 51)(15 54 29 21)(16 24 30 49)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 56 29 52)(26 49 30 53)(27 50 31 54)(28 51 32 55)(41 63 45 59)(42 64 46 60)(43 57 47 61)(44 58 48 62)```

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

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

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

Matrix representation of C4.192+ 1+4 in GL8(𝔽17)

 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 16 16 2 0 0 0 0 1 0 16 1 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
,
 5 5 0 0 0 0 0 0 12 5 0 0 0 0 0 0 12 5 10 7 0 0 0 0 12 0 5 0 0 0 0 0 0 0 0 0 8 9 9 8 0 0 0 0 8 8 9 9 0 0 0 0 9 8 9 8 0 0 0 0 9 9 9 9
,
 0 0 1 0 0 0 0 0 16 1 1 15 0 0 0 0 16 0 0 0 0 0 0 0 16 1 0 16 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
,
 1 16 16 2 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 5 0 12 0 0 0 0 0 0 12 0 5 0 0 0 0 12 0 12 0 0 0 0 0 0 5 0 5
,
 1 16 16 2 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 1 16 0 0 0 0 0 0 0 0 5 0 12 0 0 0 0 0 0 5 0 12 0 0 0 0 12 0 12 0 0 0 0 0 0 12 0 12

`G:=sub<GL(8,GF(17))| [0,1,1,1,0,0,0,0,16,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[5,12,12,12,0,0,0,0,5,5,5,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,8,8,9,9,0,0,0,0,9,8,8,9,0,0,0,0,9,9,9,9,0,0,0,0,8,9,8,9],[0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5],[1,0,0,16,0,0,0,0,16,0,16,0,0,0,0,0,16,1,0,1,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,5,0,12,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12] >;`

C4.192+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{19}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.19ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1936);`
`// by ID`

`G=gap.SmallGroup(128,1936);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,248,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=1,b^4=c^2=e^2=a^2,d^2=a^-1*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^-1*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a*b^2*d>;`
`// generators/relations`

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