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## G = C4.152+ 1+4order 128 = 27

### 15th 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 — C4.152+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C4.152+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.152+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.152+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.152+ 1+4

Generators and relations for C4.152+ 1+4
G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >

Subgroups: 476 in 205 conjugacy classes, 84 normal (30 characteristic)
C1, C2, 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, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C22×D4, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, Q8⋊D4, C22⋊SD16, C88D4, C8⋊D4, C23.19D4, C23.20D4, C22.29C24, C23.38C23, C4.152+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○SD16, C4.152+ 1+4

Character table of C4.152+ 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 0 0 2√-2 -2√-2 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 0 -2√-2 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 -2√-2 0 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 2√-2 0 0 0 0 complex lifted from D4○SD16

Smallest permutation representation of C4.152+ 1+4
On 32 points
Generators in S32
```(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(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)
(2 29)(4 31)(6 25)(8 27)(9 13)(10 20)(11 15)(12 22)(14 24)(16 18)(17 21)(19 23)
(1 23 28 9)(2 12 29 18)(3 21 30 15)(4 10 31 24)(5 19 32 13)(6 16 25 22)(7 17 26 11)(8 14 27 20)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)```

`G:=sub<Sym(32)| (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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), (2,29)(4,31)(6,25)(8,27)(9,13)(10,20)(11,15)(12,22)(14,24)(16,18)(17,21)(19,23), (1,23,28,9)(2,12,29,18)(3,21,30,15)(4,10,31,24)(5,19,32,13)(6,16,25,22)(7,17,26,11)(8,14,27,20), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)>;`

`G:=Group( (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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), (2,29)(4,31)(6,25)(8,27)(9,13)(10,20)(11,15)(12,22)(14,24)(16,18)(17,21)(19,23), (1,23,28,9)(2,12,29,18)(3,21,30,15)(4,10,31,24)(5,19,32,13)(6,16,25,22)(7,17,26,11)(8,14,27,20), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31) );`

`G=PermutationGroup([[(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(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)], [(2,29),(4,31),(6,25),(8,27),(9,13),(10,20),(11,15),(12,22),(14,24),(16,18),(17,21),(19,23)], [(1,23,28,9),(2,12,29,18),(3,21,30,15),(4,10,31,24),(5,19,32,13),(6,16,25,22),(7,17,26,11),(8,14,27,20)], [(1,11,5,15),(2,12,6,16),(3,13,7,9),(4,14,8,10),(17,32,21,28),(18,25,22,29),(19,26,23,30),(20,27,24,31)]])`

Matrix representation of C4.152+ 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
,
 0 0 10 7 0 0 0 0 5 0 0 7 0 0 0 0 0 12 5 12 0 0 0 0 12 5 5 12 0 0 0 0 0 0 0 0 7 0 2 2 0 0 0 0 3 10 13 15 0 0 0 0 8 8 7 0 0 0 0 0 1 9 3 10
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 16 0 0 0 0 0 1 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 10 10 16 0 0 0 0 0 0 7 0 16
,
 1 0 0 15 0 0 0 0 0 0 1 16 0 0 0 0 1 16 0 16 0 0 0 0 1 0 0 16 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 2 1 0 0 0 0 0 0 10 0 16 16 0 0 0 0 14 7 2 1
,
 1 0 15 0 0 0 0 0 0 0 16 1 0 0 0 0 1 0 16 0 0 0 0 0 1 16 16 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 16 0 0 0 0 0 0 10 0 16 16 0 0 0 0 0 10 0 1

`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],[0,5,0,12,0,0,0,0,0,0,12,5,0,0,0,0,10,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,0,0,0,0,7,3,8,1,0,0,0,0,0,10,8,9,0,0,0,0,2,13,7,3,0,0,0,0,2,15,0,10],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,10,7,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,0,16,2,10,14,0,0,0,0,16,1,0,7,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,1,16,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;`

C4.152+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{15}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.15ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1932);`
`// by ID`

`G=gap.SmallGroup(128,1932);`
`# by ID`

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

`G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=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=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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