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## G = C4.2- 1+4order 128 = 27

### 13rd non-split extension by C4 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.2- 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.36C24 — C4.2- 1+4
 Lower central C1 — C2 — C2×C4 — C4.2- 1+4
 Upper central C1 — C22 — C42⋊C2 — C4.2- 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.2- 1+4

Generators and relations for C4.2- 1+4
G = < a,b,c,d,e | a4=b4=c2=1, d2=ab2, e2=b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab, ebe-1=a2b, cd=dc, ece-1=a2c, ede-1=a2b2d >

Subgroups: 388 in 179 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C422C2, C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C42.6C22, C4⋊D8, C4⋊SD16, D4.2D4, C88D4, C87D4, C8⋊D4, C82D4, D4⋊Q8, D42Q8, D4.Q8, Q8.Q8, C22.34C24, C22.36C24, C4.2- 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○D8, D4○SD16, C4.2- 1+4

Character table of C4.2- 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 8 8 8 2 2 4 4 4 4 4 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 0 0 0 -2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 0 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 0 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 0 0 0 -2 -2 -2 2 -2 2 -2 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 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 orthogonal lifted from D4○D8 ρ22 4 -4 4 -4 0 0 0 0 4 -4 0 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 orthogonal lifted from D4○D8 ρ24 4 -4 4 -4 0 0 0 0 -4 4 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 ρ25 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 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 C4.2- 1+4
On 64 points
Generators in S64
```(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)(49 60 53 64)(50 61 54 57)(51 62 55 58)(52 63 56 59)
(1 28 18 43)(2 42 19 27)(3 26 20 41)(4 48 21 25)(5 32 22 47)(6 46 23 31)(7 30 24 45)(8 44 17 29)(9 58 37 49)(10 56 38 57)(11 64 39 55)(12 54 40 63)(13 62 33 53)(14 52 34 61)(15 60 35 51)(16 50 36 59)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 18 54)(2 51 19 60)(3 57 20 56)(4 53 21 62)(5 59 22 50)(6 55 23 64)(7 61 24 52)(8 49 17 58)(9 25 37 48)(10 45 38 30)(11 27 39 42)(12 47 40 32)(13 29 33 44)(14 41 34 26)(15 31 35 46)(16 43 36 28)```

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

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

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

Matrix representation of C4.2- 1+4 in GL8(𝔽17)

 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 16 0 1 0 0 0 0 1 16 16 0 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 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 2 0 0 0 0 0 0 16 1 0 0 0 0 0 16 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 15 0 0 0 0 0 0 1 0 15 0 0 0 0 1 0 16 0 0 0 0 0 0 1 0 16
,
 1 0 15 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 0 0 1 0 15 0 0 0 0 0 0 1 0 15 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 11 0 6 6 0 0 0 0 14 0 6 0 0 0 0 0 0 14 3 3 0 0 0 0 14 14 3 3 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 1 10
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 1 0 0 0 0 0 16 0 0 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

`G:=sub<GL(8,GF(17))| [1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,15,0,16],[11,14,0,14,0,0,0,0,0,0,14,14,0,0,0,0,6,6,3,3,0,0,0,0,6,0,3,3,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10],[16,0,16,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0] >;`

C4.2- 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,1989);`
`// by ID`

`G=gap.SmallGroup(128,1989);`
`# by ID`

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

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

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