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## G = C4.162+ 1+4order 128 = 27

### 16th 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 — C4.162+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.38C23 — C4.162+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.162+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.162+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.162+ 1+4

Generators and relations for C4.162+ 1+4
G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece=a2c, ede=a-1b2d >

Subgroups: 404 in 195 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, Q8⋊D4, C22⋊Q16, D4.7D4, C88D4, C8.18D4, C8⋊D4, C8.D4, C23.19D4, C23.47D4, C23.48D4, C23.20D4, C23.38C23, C22.31C24, C4.162+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○SD16, Q8○D8, C4.162+ 1+4

Character table of C4.162+ 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 4 4 8 2 2 4 4 4 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 0 -2 -2 2 2 2 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 0 -2 -2 -2 2 -2 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 0 -2 -2 2 -2 -2 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 0 -2 -2 -2 -2 2 0 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 -4 4 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 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 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 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 2√-2 0 -2√-2 0 0 0 complex lifted from D4○SD16

Smallest permutation representation of C4.162+ 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 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 48 37 44)(34 41 38 45)(35 42 39 46)(36 43 40 47)
(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 55)(2 19)(3 49)(4 21)(5 51)(6 23)(7 53)(8 17)(9 34)(10 44)(11 36)(12 46)(13 38)(14 48)(15 40)(16 42)(18 57)(20 59)(22 61)(24 63)(25 37)(26 47)(27 39)(28 41)(29 33)(30 43)(31 35)(32 45)(50 60)(52 62)(54 64)(56 58)
(1 62 57 6)(2 5 58 61)(3 60 59 4)(7 64 63 8)(9 12 30 25)(10 32 31 11)(13 16 26 29)(14 28 27 15)(17 49 54 20)(18 19 55 56)(21 53 50 24)(22 23 51 52)(33 34 42 43)(35 40 44 41)(36 48 45 39)(37 38 46 47)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)```

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

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

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

Matrix representation of C4.162+ 1+4 in GL8(𝔽17)

 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 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 7 0 4 0 0 0 0 10 0 13 0 0 0 0 0 0 13 0 10 0 0 0 0 4 0 7 0 0 0 0 0 0 0 0 0 14 14 0 0 0 0 0 0 3 14 0 0 0 0 0 0 0 0 14 14 0 0 0 0 0 0 3 14
,
 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 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 7 0 4 0 0 0 0 7 0 4 0 0 0 0 0 0 13 0 10 0 0 0 0 13 0 10 0 0 0 0 0 0 0 0 0 3 14 0 0 0 0 0 0 14 14 0 0 0 0 0 0 0 0 14 3 0 0 0 0 0 0 3 3
,
 0 13 0 10 0 0 0 0 13 0 10 0 0 0 0 0 0 7 0 4 0 0 0 0 7 0 4 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 16 0 0 10 0 0 0 0 7 0 0 16 0 0 0 0 0 7 1 0

`G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,10,0,4,0,0,0,0,7,0,13,0,0,0,0,0,0,13,0,7,0,0,0,0,4,0,10,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,14,14],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,7,0,13,0,0,0,0,7,0,13,0,0,0,0,0,0,4,0,10,0,0,0,0,4,0,10,0,0,0,0,0,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3],[0,13,0,7,0,0,0,0,13,0,7,0,0,0,0,0,0,10,0,4,0,0,0,0,10,0,4,0,0,0,0,0,0,0,0,0,0,16,7,0,0,0,0,0,1,0,0,7,0,0,0,0,10,0,0,1,0,0,0,0,0,10,16,0] >;`

C4.162+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{16}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.16ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1933);`
`// by ID`

`G=gap.SmallGroup(128,1933);`
`# by ID`

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

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

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