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G = 2- 1+4⋊3C6order 192 = 26·3

2nd semidirect product of 2- 1+4 and C6 acting via C6/C2=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — 2- 1+4⋊3C6
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×C4.A4 — 2- 1+4⋊3C6
 Lower central Q8 — 2- 1+4⋊3C6
 Upper central C1 — C4 — C4○D4

Generators and relations for 2- 1+43C6
G = < a,b,c,d,e | a4=b2=e6=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a2b, dcd-1=a2c, ece-1=cd, ede-1=c >

Subgroups: 573 in 195 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C2×C6, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, C2×SL2(𝔽3), C4.A4, C4.A4, C3×C4○D4, C2.C25, C2×C4.A4, Q8.A4, D4.A4, 2- 1+43C6
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C22×A4, C23×A4, 2- 1+43C6

Smallest permutation representation of 2- 1+43C6
On 32 points
Generators in S32
```(1 2 4 3)(5 8 6 7)(9 14 19 16)(10 12 20 17)(11 13 18 15)(21 27 24 30)(22 28 25 31)(23 29 26 32)
(1 7)(2 6)(3 5)(4 8)(9 31)(10 29)(11 27)(12 23)(13 21)(14 25)(15 24)(16 22)(17 26)(18 30)(19 28)(20 32)
(1 20 4 10)(2 17 3 12)(5 23 6 26)(7 32 8 29)(9 11 19 18)(13 16 15 14)(21 22 24 25)(27 28 30 31)
(1 18 4 11)(2 15 3 13)(5 21 6 24)(7 30 8 27)(9 20 19 10)(12 14 17 16)(22 23 25 26)(28 29 31 32)
(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)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C ··· 6H 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 2 2 2 6 6 6 6 4 4 1 1 2 2 2 6 6 6 6 4 4 8 ··· 8 4 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 A4 C2×A4 C2×A4 C2×A4 2- 1+4⋊3C6 kernel 2- 1+4⋊3C6 C2×C4.A4 Q8.A4 D4.A4 C2.C25 C2×C4○D4 2+ 1+4 2- 1+4 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 1 3 2 6 2 6 1 3 3 1 6

Matrix representation of 2- 1+43C6 in GL4(𝔽5) generated by

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

2- 1+43C6 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes_3C_6`
`% in TeX`

`G:=Group("ES-(2,2):3C6");`
`// GroupNames label`

`G:=SmallGroup(192,1504);`
`// by ID`

`G=gap.SmallGroup(192,1504);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,235,172,404,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=e^6=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b,d*c*d^-1=a^2*c,e*c*e^-1=c*d,e*d*e^-1=c>;`
`// generators/relations`

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