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## G = 3- 1+2⋊A4order 324 = 22·34

### 1st semidirect product of 3- 1+2 and A4 acting via A4/C22=C3

Aliases: C62.5C32, 3- 1+21A4, (C2×C6).9He3, C32.A44C3, C32⋊A4.2C3, C32.5(C3×A4), C3.10(C32⋊A4), C223(He3.C3), (C22×3- 1+2)⋊1C3, (C3×C3.A4)⋊4C3, SmallGroup(324,57)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — 3- 1+2⋊A4
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — 3- 1+2⋊A4
 Lower central C22 — C2×C6 — C62 — 3- 1+2⋊A4
 Upper central C1 — C3 — C32 — 3- 1+2

Generators and relations for 3- 1+2⋊A4
G = < a,b,c,d,e | a9=b3=c2=d2=e3=1, bab-1=a4, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a3b, ece-1=cd=dc, ede-1=c >

Character table of 3- 1+2⋊A4

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 18A 18B 18C 18D 18E 18F size 1 3 1 1 3 3 36 36 3 3 9 9 9 9 12 12 12 12 12 12 36 36 9 9 9 9 9 9 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 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ8 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ9 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ10 3 -1 3 3 3 3 0 0 -1 -1 -1 -1 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 2 -1+√-3 -1-√-3 2 -1+√-3 -1-√-3 complex lifted from C32⋊A4 ρ12 3 -1 3 3 3 3 0 0 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 ζ65 ζ6 ζ65 ζ6 ζ65 ζ6 complex lifted from C3×A4 ρ13 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 -1-√-3 -1-√-3 -1+√-3 -1+√-3 2 2 complex lifted from C32⋊A4 ρ14 3 -1 3 3 3 3 0 0 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 ζ6 ζ65 ζ6 ζ65 ζ6 ζ65 complex lifted from C3×A4 ρ15 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 2 -1-√-3 -1+√-3 2 -1-√-3 -1+√-3 complex lifted from C32⋊A4 ρ16 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ17 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 -1-√-3 2 2 -1+√-3 -1+√-3 -1-√-3 complex lifted from C32⋊A4 ρ18 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -1 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 -1+√-3 -1+√-3 -1-√-3 -1-√-3 2 2 complex lifted from C32⋊A4 ρ19 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ20 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -1 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 -1+√-3 2 2 -1-√-3 -1-√-3 -1+√-3 complex lifted from C32⋊A4 ρ21 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 2ζ97+ζ9 ζ98+2ζ92 2ζ95+ζ92 2ζ98+ζ95 ζ94+2ζ9 ζ97+2ζ94 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ22 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ94+2ζ9 2ζ98+ζ95 ζ98+2ζ92 2ζ95+ζ92 ζ97+2ζ94 2ζ97+ζ9 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ23 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 2ζ95+ζ92 ζ97+2ζ94 ζ94+2ζ9 2ζ97+ζ9 ζ98+2ζ92 2ζ98+ζ95 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ24 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ97+2ζ94 2ζ95+ζ92 2ζ98+ζ95 ζ98+2ζ92 2ζ97+ζ9 ζ94+2ζ9 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ25 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 2ζ98+ζ95 ζ94+2ζ9 2ζ97+ζ9 ζ97+2ζ94 2ζ95+ζ92 ζ98+2ζ92 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ26 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ98+2ζ92 2ζ97+ζ9 ζ97+2ζ94 ζ94+2ζ9 2ζ98+ζ95 2ζ95+ζ92 0 0 0 0 0 0 0 0 complex lifted from He3.C3 ρ27 9 -3 -9-9√-3/2 -9+9√-3/2 0 0 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 9 -3 -9+9√-3/2 -9-9√-3/2 0 0 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of 3- 1+2⋊A4
On 54 points
Generators in S54
```(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 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(20 26 23)(21 24 27)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 49 52)(48 54 51)
(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 46)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)
(1 49 34)(2 47 32)(3 51 33)(4 52 28)(5 50 35)(6 54 36)(7 46 31)(8 53 29)(9 48 30)(10 44 26)(11 42 24)(12 37 25)(13 38 20)(14 45 27)(15 40 19)(16 41 23)(17 39 21)(18 43 22)```

`G:=sub<Sym(54)| (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51), (10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,46)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,49,34)(2,47,32)(3,51,33)(4,52,28)(5,50,35)(6,54,36)(7,46,31)(8,53,29)(9,48,30)(10,44,26)(11,42,24)(12,37,25)(13,38,20)(14,45,27)(15,40,19)(16,41,23)(17,39,21)(18,43,22)>;`

`G:=Group( (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51), (10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,46)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,49,34)(2,47,32)(3,51,33)(4,52,28)(5,50,35)(6,54,36)(7,46,31)(8,53,29)(9,48,30)(10,44,26)(11,42,24)(12,37,25)(13,38,20)(14,45,27)(15,40,19)(16,41,23)(17,39,21)(18,43,22) );`

`G=PermutationGroup([[(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(20,26,23),(21,24,27),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,49,52),(48,54,51)], [(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,46),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(1,49,34),(2,47,32),(3,51,33),(4,52,28),(5,50,35),(6,54,36),(7,46,31),(8,53,29),(9,48,30),(10,44,26),(11,42,24),(12,37,25),(13,38,20),(14,45,27),(15,40,19),(16,41,23),(17,39,21),(18,43,22)]])`

Matrix representation of 3- 1+2⋊A4 in GL6(𝔽19)

 0 1 0 0 0 0 8 18 4 0 0 0 18 12 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 7 0 0 0 0 1 18 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 11 18 0 0 0 0 7 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 12 0 1
,
 14 3 7 0 0 0 5 0 0 0 0 0 16 0 5 0 0 0 0 0 0 11 17 0 0 0 0 0 8 1 0 0 0 0 12 0

`G:=sub<GL(6,GF(19))| [0,8,18,0,0,0,1,18,12,0,0,0,0,4,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,1,0,0,0,0,7,18,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,7,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,12,0,0,0,0,18,0,0,0,0,0,0,1],[14,5,16,0,0,0,3,0,0,0,0,0,7,0,5,0,0,0,0,0,0,11,0,0,0,0,0,17,8,12,0,0,0,0,1,0] >;`

3- 1+2⋊A4 in GAP, Magma, Sage, TeX

`3_-^{1+2}\rtimes A_4`
`% in TeX`

`G:=Group("ES-(3,1):A4");`
`// GroupNames label`

`G:=SmallGroup(324,57);`
`// by ID`

`G=gap.SmallGroup(324,57);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,115,1136,224,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^2=e^3=1,b*a*b^-1=a^4,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;`
`// generators/relations`

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