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## G = C32⋊2+ 1+4order 288 = 25·32

### The semidirect product of C32 and 2+ 1+4 acting via 2+ 1+4/C23=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊2+ 1+4
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×C3⋊D4 — C32⋊2+ 1+4
 Lower central C32 — C3×C6 — C32⋊2+ 1+4
 Upper central C1 — C2 — C23

Generators and relations for C32⋊2+ 1+4
G = < a,b,c,d,e,f | a3=b3=c4=d2=f2=1, e2=c2, ab=ba, cac-1=dad=a-1, ae=ea, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 1410 in 359 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, 2+ 1+4, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C327D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×C62, D46D6, D6.3D6, D6.4D6, S3×C3⋊D4, Dic3⋊D6, C6×C3⋊D4, C2×C327D4, C32⋊2+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D46D6, C22×S32, C32⋊2+ 1+4

Permutation representations of C32⋊2+ 1+4
On 24 points - transitive group 24T582
Generators in S24
```(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
(1 22 3 24)(2 23 4 21)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)```

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

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

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

`G:=TransitiveGroup(24,582);`

On 24 points - transitive group 24T609
Generators in S24
```(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 10 14)(6 15 11)(7 12 16)(8 13 9)
(1 21 20)(2 22 17)(3 23 18)(4 24 19)(5 14 10)(6 15 11)(7 16 12)(8 13 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
(1 2 3 4)(5 8 7 6)(9 16 11 14)(10 13 12 15)(17 23 19 21)(18 24 20 22)
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)```

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

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

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

`G:=TransitiveGroup(24,609);`

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6Q 6R 6S 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 2 2 2 6 6 6 6 18 18 2 2 4 6 6 6 6 18 18 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2+ 1+4 S32 C2×S32 D4⋊6D6 C32⋊2+ 1+4 kernel C32⋊2+ 1+4 D6.3D6 D6.4D6 S3×C3⋊D4 Dic3⋊D6 C6×C3⋊D4 C2×C32⋊7D4 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C32 C23 C22 C3 C1 # reps 1 4 2 4 2 2 1 2 2 8 2 2 1 1 3 4 4

Matrix representation of C32⋊2+ 1+4 in GL4(𝔽7) generated by

 4 1 2 1 3 3 6 4 6 6 4 2 6 1 4 1
,
 6 0 3 1 5 3 6 0 1 6 4 5 3 3 2 6
,
 4 6 0 0 3 3 0 0 3 2 1 4 0 4 3 6
,
 1 4 2 0 5 3 3 1 4 6 0 5 3 3 2 3
,
 4 3 3 4 3 6 4 1 3 0 3 4 0 3 4 1
,
 3 0 3 1 5 0 6 0 1 6 1 5 3 3 2 3
`G:=sub<GL(4,GF(7))| [4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,3,3,0,6,3,2,4,0,0,1,3,0,0,4,6],[1,5,4,3,4,3,6,3,2,3,0,2,0,1,5,3],[4,3,3,0,3,6,0,3,3,4,3,4,4,1,4,1],[3,5,1,3,0,0,6,3,3,6,1,2,1,0,5,3] >;`

C32⋊2+ 1+4 in GAP, Magma, Sage, TeX

`C_3^2\rtimes 2_+^{1+4}`
`% in TeX`

`G:=Group("C3^2:ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(288,978);`
`// by ID`

`G=gap.SmallGroup(288,978);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,1356,9414]);`
`// Polycyclic`

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

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