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## G = C24.3C22order 64 = 26

### 3rd non-split extension by C24 of C22 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.3C22
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C24.3C22
 Lower central C1 — C22 — C24.3C22
 Upper central C1 — C23 — C24.3C22
 Jennings C1 — C23 — C24.3C22

Generators and relations for C24.3C22
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=d, f2=c, eae-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, de=ed, df=fd >

Subgroups: 233 in 129 conjugacy classes, 53 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×10], C2×C4 [×10], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C24.3C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22

Character table of C24.3C22

 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 1 1 1 1 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 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 i -1 1 -1 1 -i i i -i i -i -i i i -i -i linear of order 4 ρ10 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -i 1 -1 1 -1 i -i i -i i i -i -i i -i i linear of order 4 ρ11 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 i -1 1 -1 1 -i i i -i i -i -i -i -i i i linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 -i 1 -1 1 -1 i -i i -i i i -i i -i i -i linear of order 4 ρ13 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 -i -1 1 -1 1 i -i -i i -i i i i i -i -i linear of order 4 ρ14 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 i 1 -1 1 -1 -i i -i i -i -i i -i i -i i linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -i -1 1 -1 1 i -i -i i -i i i -i -i i i linear of order 4 ρ16 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 i 1 -1 1 -1 -i i -i i -i -i i i -i i -i linear of order 4 ρ17 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 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 2 -2 -2 0 0 0 0 0 -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 2 -2 2 0 0 0 0 2 0 0 0 0 -2 -2 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 0 2 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ22 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 -2 -2 -2 2 2 -2 2 0 0 0 0 -2 0 0 0 0 2 2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -2i -2i 2i 0 2i 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 -2 2 2 -2 -2 -2 0 0 0 0 2i 0 0 0 0 2i -2i 0 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 2i 2i -2i 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 -2 2 2 -2 -2 -2 0 0 0 0 -2i 0 0 0 0 -2i 2i 0 0 0 2i 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C24.3C22
On 32 points
Generators in S32
```(1 14)(2 17)(3 16)(4 19)(5 30)(6 28)(7 32)(8 26)(9 13)(10 20)(11 15)(12 18)(21 29)(22 27)(23 31)(24 25)
(1 10)(2 11)(3 12)(4 9)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(13 24)(14 21)(15 22)(16 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)
(1 21 29 14)(2 15 30 22)(3 23 31 16)(4 13 32 24)(5 11 17 27)(6 28 18 12)(7 9 19 25)(8 26 20 10)```

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

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

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

Matrix representation of C24.3C22 in GL5(𝔽5)

 4 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 2 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 3 0
,
 1 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 4 0

`G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,3,0],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0] >;`

C24.3C22 in GAP, Magma, Sage, TeX

`C_2^4._3C_2^2`
`% in TeX`

`G:=Group("C2^4.3C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,71);`
`// by ID`

`G=gap.SmallGroup(64,71);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362,86]);`
`// Polycyclic`

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

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