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## G = C2×C3⋊F9order 432 = 24·33

### Direct product of C2 and C3⋊F9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C3⋊F9
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C3⋊F9 — C2×C3⋊F9
 Lower central C33 — C2×C3⋊F9
 Upper central C1 — C2

Generators and relations for C2×C3⋊F9
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, ece-1=cd=dc, ede-1=c >

Subgroups: 368 in 58 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C2×C12, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C2×C3⋊C8, C3×C3⋊S3, C32×C6, F9, C2×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C2×F9, C3⋊F9, C6×C32⋊C4, C2×C3⋊F9
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C2×Dic3, C2×C3⋊C8, F9, C2×F9, C3⋊F9, C2×C3⋊F9

Character table of C2×C3⋊F9

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D size 1 1 9 9 2 8 8 8 9 9 9 9 2 8 8 8 18 18 27 27 27 27 27 27 27 27 18 18 18 18 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 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 -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 -i i i -i -i i i -i -1 -1 -1 -1 linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 i i i -i -i -i -i i -1 1 1 -1 linear of order 4 ρ7 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 i -i -i i i -i -i i -1 -1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -i -i -i i i i i -i -1 1 1 -1 linear of order 4 ρ9 1 1 -1 -1 1 1 1 1 -i -i i i 1 1 1 1 -1 -1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 i -i i -i linear of order 8 ρ10 1 1 -1 -1 1 1 1 1 i i -i -i 1 1 1 1 -1 -1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 -i i -i i linear of order 8 ρ11 1 -1 1 -1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 1 ζ85 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 -i -i i i linear of order 8 ρ12 1 -1 1 -1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 1 ζ83 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 i i -i -i linear of order 8 ρ13 1 1 -1 -1 1 1 1 1 i i -i -i 1 1 1 1 -1 -1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 -i i -i i linear of order 8 ρ14 1 -1 1 -1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 1 ζ87 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 i i -i -i linear of order 8 ρ15 1 -1 1 -1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 1 ζ8 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 -i -i i i linear of order 8 ρ16 1 1 -1 -1 1 1 1 1 -i -i i i 1 1 1 1 -1 -1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 i -i i -i linear of order 8 ρ17 2 -2 -2 2 -1 -1 2 -1 -2 2 2 -2 1 1 -2 1 -1 1 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 -1 -1 2 -1 2 2 2 2 -1 -1 2 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 -2 -2 2 -1 -1 2 -1 2 -2 -2 2 1 1 -2 1 -1 1 0 0 0 0 0 0 0 0 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ20 2 2 2 2 -1 -1 2 -1 -2 -2 -2 -2 -1 -1 2 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ21 2 2 -2 -2 -1 -1 2 -1 2i 2i -2i -2i -1 -1 2 -1 1 1 0 0 0 0 0 0 0 0 i -i i -i complex lifted from C3⋊C8 ρ22 2 2 -2 -2 -1 -1 2 -1 -2i -2i 2i 2i -1 -1 2 -1 1 1 0 0 0 0 0 0 0 0 -i i -i i complex lifted from C3⋊C8 ρ23 2 -2 2 -2 -1 -1 2 -1 2i -2i 2i -2i 1 1 -2 1 1 -1 0 0 0 0 0 0 0 0 -i -i i i complex lifted from C3⋊C8 ρ24 2 -2 2 -2 -1 -1 2 -1 -2i 2i -2i 2i 1 1 -2 1 1 -1 0 0 0 0 0 0 0 0 i i -i -i complex lifted from C3⋊C8 ρ25 8 -8 0 0 8 -1 -1 -1 0 0 0 0 -8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×F9 ρ26 8 8 0 0 8 -1 -1 -1 0 0 0 0 8 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ27 8 8 0 0 -4 1+3√-3/2 -1 1-3√-3/2 0 0 0 0 -4 1+3√-3/2 -1 1-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3⋊F9 ρ28 8 -8 0 0 -4 1-3√-3/2 -1 1+3√-3/2 0 0 0 0 4 -1+3√-3/2 1 -1-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ29 8 8 0 0 -4 1-3√-3/2 -1 1+3√-3/2 0 0 0 0 -4 1-3√-3/2 -1 1+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3⋊F9 ρ30 8 -8 0 0 -4 1+3√-3/2 -1 1-3√-3/2 0 0 0 0 4 -1-3√-3/2 1 -1+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C2×C3⋊F9
On 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 28 34)(18 35 29)(19 30 36)(20 37 31)(21 32 38)(22 39 25)(23 26 40)(24 33 27)
(1 46 11)(2 12 47)(4 14 41)(5 15 42)(6 43 16)(8 45 10)(18 29 35)(19 30 36)(20 37 31)(22 39 25)(23 40 26)(24 27 33)
(1 46 11)(2 47 12)(3 13 48)(5 15 42)(6 16 43)(7 44 9)(17 28 34)(19 30 36)(20 31 37)(21 38 32)(23 40 26)(24 33 27)
(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)

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

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

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

Matrix representation of C2×C3⋊F9 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 64 0 0 0 3 3 0 0 0 64 0 0 39 39 0 0 0 0 64 0 31 31 0 0 0 0 0 64
,
 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 14 29 1 0 0 0 0 0 60 53 0 1 0 0 0 0 0 0 0 0 64 0 0 0 0 70 0 0 8 8 0 0 39 0 0 0 0 0 64 0 0 42 0 0 3 0 0 8
,
 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 42 8 0 0 0 0 0 31 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 27 24 0 0 0 1 0 0 0 34 0 0 46 0 8 0 31 0 0 0 24 0 0 64
,
 27 27 0 0 1 66 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 59 0 72 0 0 0 0 0 13 72 0 60 14 0 66 0 0 0 0 0 0 72 72 0 46 0 0 0 0 0 46 0 14 0 0 0 0 0 27 0 13 0 0

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[8,0,0,0,0,3,39,31,0,8,0,0,0,3,39,31,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[8,0,14,60,0,0,39,0,0,64,29,53,0,70,0,42,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,8,0,3,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8],[8,0,0,31,0,27,0,31,0,64,42,0,0,24,34,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,46,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[27,0,0,0,60,0,0,0,27,0,0,0,14,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,66,72,46,27,1,72,0,0,0,0,0,0,66,0,59,13,0,46,14,13,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0] >;

C2×C3⋊F9 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes F_9
% in TeX

G:=Group("C2xC3:F9");
// GroupNames label

G:=SmallGroup(432,752);
// by ID

G=gap.SmallGroup(432,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,2244,718,165,677,691,530,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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