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## G = C3×C23.A4order 288 = 25·32

### Direct product of C3 and C23.A4

Aliases: C3×C23.A4, (C4×C12)⋊5C6, C42⋊C33C6, C41D4⋊C32, C422(C3×C6), C23.2(C3×A4), C22.4(C6×A4), (C22×C6).6A4, (C3×C41D4)⋊C3, (C3×C42⋊C3)⋊7C2, (C2×C6).12(C2×A4), SmallGroup(288,636)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C3×C23.A4
 Chief series C1 — C22 — C42 — C4×C12 — C3×C42⋊C3 — C3×C23.A4
 Lower central C42 — C3×C23.A4
 Upper central C1 — C3

Generators and relations for C3×C23.A4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, bd=db, ebe-1=bcd, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >

Subgroups: 348 in 68 conjugacy classes, 18 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C22, C6, C2×C4, D4, C23, C23, C32, C12, A4, C2×C6, C2×C6, C42, C2×D4, C3×C6, C2×C12, C3×D4, C2×A4, C22×C6, C22×C6, C41D4, C3×A4, C42⋊C3, C4×C12, C6×D4, C6×A4, C23.A4, C3×C41D4, C3×C42⋊C3, C3×C23.A4
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C3×A4, C6×A4, C23.A4, C3×C23.A4

Character table of C3×C23.A4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B 12C 12D size 1 3 4 12 1 1 16 16 16 16 16 16 6 6 3 3 4 4 12 12 16 16 16 16 16 16 6 6 6 6 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 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 1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 -1 -1 -1 -1 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 1 1 linear of order 6 ρ4 1 1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 ζ3 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 ζ65 ζ6 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ5 1 1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 ζ32 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 ζ6 ζ65 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ7 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ8 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ10 1 1 -1 -1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 -1 -1 -1 -1 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 1 1 linear of order 6 ρ11 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ12 1 1 -1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 -1 ζ65 -1 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ13 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ14 1 1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ15 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ16 1 1 -1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 -1 ζ6 -1 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ17 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ18 1 1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ19 3 3 -3 1 3 3 0 0 0 0 0 0 -1 -1 3 3 -3 -3 1 1 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ20 3 3 3 -1 3 3 0 0 0 0 0 0 -1 -1 3 3 3 3 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from A4 ρ21 3 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 0 0 0 0 0 0 ζ65 ζ6 ζ65 ζ6 complex lifted from C3×A4 ρ22 3 3 -3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 3+3√-3/2 3-3√-3/2 ζ32 ζ3 0 0 0 0 0 0 ζ65 ζ6 ζ65 ζ6 complex lifted from C6×A4 ρ23 3 3 -3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 3-3√-3/2 3+3√-3/2 ζ3 ζ32 0 0 0 0 0 0 ζ6 ζ65 ζ6 ζ65 complex lifted from C6×A4 ρ24 3 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 0 0 0 0 0 0 ζ6 ζ65 ζ6 ζ65 complex lifted from C3×A4 ρ25 6 -2 0 0 6 6 0 0 0 0 0 0 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2 -2 2 2 orthogonal lifted from C23.A4 ρ26 6 -2 0 0 6 6 0 0 0 0 0 0 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 orthogonal lifted from C23.A4 ρ27 6 -2 0 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 2 -2 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 1+√-3 1-√-3 -1-√-3 -1+√-3 complex faithful ρ28 6 -2 0 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 2 -2 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 1-√-3 1+√-3 -1+√-3 -1-√-3 complex faithful ρ29 6 -2 0 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 -2 2 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 -1+√-3 -1-√-3 1-√-3 1+√-3 complex faithful ρ30 6 -2 0 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 -2 2 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 -1-√-3 -1+√-3 1+√-3 1-√-3 complex faithful

Smallest permutation representation of C3×C23.A4
On 36 points
Generators in S36
(1 9 5)(2 8 4)(3 11 7)(6 12 10)(13 18 21)(14 19 22)(15 20 23)(16 17 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)
(1 12)(2 11)(3 4)(5 6)(7 8)(9 10)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(1 4)(2 9)(3 12)(5 8)(6 7)(10 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 4)(2 9)(3 12)(5 8)(6 7)(10 11)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 3 4 12)(2 10 9 11)(5 7 8 6)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(1 26 14)(2 36 17)(3 25 15)(4 28 16)(5 30 22)(6 31 21)(7 29 23)(8 32 24)(9 34 19)(10 35 18)(11 33 20)(12 27 13)

G:=sub<Sym(36)| (1,9,5)(2,8,4)(3,11,7)(6,12,10)(13,18,21)(14,19,22)(15,20,23)(16,17,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,12)(2,11)(3,4)(5,6)(7,8)(9,10)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,4)(2,9)(3,12)(5,8)(6,7)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,4)(2,9)(3,12)(5,8)(6,7)(10,11)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,3,4,12)(2,10,9,11)(5,7,8,6)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (1,26,14)(2,36,17)(3,25,15)(4,28,16)(5,30,22)(6,31,21)(7,29,23)(8,32,24)(9,34,19)(10,35,18)(11,33,20)(12,27,13)>;

G:=Group( (1,9,5)(2,8,4)(3,11,7)(6,12,10)(13,18,21)(14,19,22)(15,20,23)(16,17,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,12)(2,11)(3,4)(5,6)(7,8)(9,10)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,4)(2,9)(3,12)(5,8)(6,7)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,4)(2,9)(3,12)(5,8)(6,7)(10,11)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,3,4,12)(2,10,9,11)(5,7,8,6)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (1,26,14)(2,36,17)(3,25,15)(4,28,16)(5,30,22)(6,31,21)(7,29,23)(8,32,24)(9,34,19)(10,35,18)(11,33,20)(12,27,13) );

G=PermutationGroup([[(1,9,5),(2,8,4),(3,11,7),(6,12,10),(13,18,21),(14,19,22),(15,20,23),(16,17,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32)], [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(1,4),(2,9),(3,12),(5,8),(6,7),(10,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,4),(2,9),(3,12),(5,8),(6,7),(10,11),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36)], [(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,3,4,12),(2,10,9,11),(5,7,8,6),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36)], [(1,26,14),(2,36,17),(3,25,15),(4,28,16),(5,30,22),(6,31,21),(7,29,23),(8,32,24),(9,34,19),(10,35,18),(11,33,20),(12,27,13)]])

Matrix representation of C3×C23.A4 in GL9(𝔽13)

 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 1 0 0 0 0 12 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 12 0 0 0 0 0 0 1 0 0 12 0 0 0 0 0 1 0 0 0 12 0 0 0 0 1 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 1 1 0 0 0 0 0 0 12 1 0 1 0 0 0 0 0 12 1 0 0 0 1 0 0 0 0 1 0 0 12 0
,
 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 12 0 0 0 0 0 12 1 1 0 0 0 0 0 0 0 1 0 0 12 0 0 0 0 0 1 0 0 0 12
,
 0 4 0 0 0 0 0 0 0 9 4 5 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0 0 0 0 0 0 1 0 12 0 0 0 0 0 0 0 1 12 0

G:=sub<GL(9,GF(13))| [3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,12,12,12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,12,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,12,12,12,12,0,0,0,0,2,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0],[1,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,12,0,12,0,0,0,0,0,2,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[0,9,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,11,12,12,12,12,12,0,0,0,0,1,0,0,0,0] >;

C3×C23.A4 in GAP, Magma, Sage, TeX

C_3\times C_2^3.A_4
% in TeX

G:=Group("C3xC2^3.A4");
// GroupNames label

G:=SmallGroup(288,636);
// by ID

G=gap.SmallGroup(288,636);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,6555,514,360,3784,3476,102,3036,5305]);
// Polycyclic

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

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