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## G = C22.19C24order 64 = 26

### 5th central stem extension by C22 of C24

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.19C24
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24
 Lower central C1 — C22 — C22.19C24
 Upper central C1 — C2×C4 — C22.19C24
 Jennings C1 — C22 — C22.19C24

Generators and relations for C22.19C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 249 in 165 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C22.19C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24

Character table of C22.19C24

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 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 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 -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 -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 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 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 -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 -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 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 -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 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 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 -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 -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 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 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 -1 -1 linear of order 2 ρ17 2 2 -2 -2 0 0 -2 0 2 0 0 0 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 0 0 2 0 -2 0 0 0 -2 2 -2 2 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 0 -2 0 2 0 0 0 -2 2 -2 2 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 0 0 2 0 -2 0 0 0 2 -2 2 -2 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 2 -2 0 -2 0 2 0 0 0 0 2i 2i -2i -2i 0 2i 0 0 -2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 2 0 0 0 0 -2 0 0 2i -2i -2i 2i -2i 0 0 0 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 -2 2 -2 0 2 0 -2 0 0 0 0 2i 2i -2i -2i 0 -2i 0 0 2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 -2 2 2 0 0 0 0 -2 0 0 -2i 2i 2i -2i 2i 0 0 0 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 -2 -2 2 -2 0 0 0 0 2 0 0 -2i 2i 2i -2i -2i 0 0 0 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 -2 0 0 0 0 2 0 0 2i -2i -2i 2i 2i 0 0 0 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 2 -2 0 -2 0 2 0 0 0 0 -2i -2i 2i 2i 0 -2i 0 0 2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 2 -2 0 2 0 -2 0 0 0 0 -2i -2i 2i 2i 0 2i 0 0 -2i 0 0 0 0 0 0 0 complex lifted from C4○D4

Permutation representations of C22.19C24
On 16 points - transitive group 16T117
Generators in S16
```(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)```

`G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;`

`G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );`

`G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,3),(2,4),(5,15),(6,16),(7,13),(8,14),(9,11),(10,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])`

`G:=TransitiveGroup(16,117);`

Matrix representation of C22.19C24 in GL4(𝔽5) generated by

 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 1
,
 1 0 0 0 0 4 0 0 0 0 4 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 3 0 0 0 0 3
`G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3] >;`

C22.19C24 in GAP, Magma, Sage, TeX

`C_2^2._{19}C_2^4`
`% in TeX`

`G:=Group("C2^2.19C2^4");`
`// GroupNames label`

`G:=SmallGroup(64,206);`
`// by ID`

`G=gap.SmallGroup(64,206);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,217,650,69]);`
`// Polycyclic`

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

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