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G = C3×C2.F9  order 432 = 24·33

Direct product of C3 and C2.F9

direct product, metabelian, soluble, monomial, A-group

Aliases: C3×C2.F9, C32⋊C48, C6.4F9, C33⋊1C16, C2.(C3×F9), (C3×C6).C24, (C32×C6).1C8, C3⋊Dic3.2C12, C32⋊2C8.1C6, (C3×C3⋊Dic3).1C4, (C3×C32⋊2C8).1C2, SmallGroup(432,565)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C2.F9
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C3×C32⋊2C8 — C3×C2.F9
Lower central
C32 — C3×C2.F9
Upper central
C1 — C6

Generators and relations for C3×C2.F9
 G = < a,b,c,d,e | a3=b2=c3=d3=1, e8=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

C1
C2
C3
4C3
8C3
9C4
C6
4C6
8C6
C32
4C32
8C32
9C8
9C12
12Dic3
C3×C6
4C3×C6
8C3×C6
C33
9C16
9C24
C3⋊Dic3
12C3×Dic3
C32×C6
9C48
C32⋊2C8
C3×C3⋊Dic3
C2.F9
C3×C32⋊2C8
C3×C2.F9

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

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

(2 44 19)(3 45 20)(4 21 46)(6 23 48)(7 24 33)(8 34 25)(10 36 27)(11 37 28)(12 29 38)(14 31 40)(15 32 41)(16 42 17)

(1 43 18)(3 45 20)(4 46 21)(5 22 47)(7 24 33)(8 25 34)(9 35 26)(11 37 28)(12 38 29)(13 30 39)(15 32 41)(16 17 42)

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

54 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D12A12B12C12D16A···16H24A···24H48A···48P
order1233333446666688881212121216···1624···2448···48
size11118889911888999999999···99···99···9

54 irreducible representations

dim11111111118888
type+++-
imageC1C2C3C4C6C8C12C16C24C48F9C2.F9C3×F9C3×C2.F9
kernelC3×C2.F9C3×C32⋊2C8C2.F9C3×C3⋊Dic3C32⋊2C8C32×C6C3⋊Dic3C33C3×C6C32C6C3C2C1
# reps112224488161122

Matrix representation of C3×C2.F9 ►in GL9(𝔽97)

6100000000
0610000000
0061000000
0006100000
0000610000
0000061000
0000006100
0000000610
0000000061
,
9600000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
0006100000
0000350000
0000035000
0000006100
0000000610
091221400115835
,
100000000
0350000000
0061000000
0006100000
0000350000
0000061000
0000003500
000000010
0681922788901
,
2700000000
000001000
000000100
000000010
0912219708685134
000100000
000010000
001000000
078934400027

G:=sub<GL(9,GF(97))| [61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,0,0,0,61],[96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,91,0,0,1,0,0,0,0,0,22,0,0,0,61,0,0,0,0,14,0,0,0,0,35,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,61,0,11,0,0,0,0,0,0,0,61,58,0,0,0,0,0,0,0,0,35],[1,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,0,6,0,0,61,0,0,0,0,0,81,0,0,0,61,0,0,0,0,92,0,0,0,0,35,0,0,0,27,0,0,0,0,0,61,0,0,8,0,0,0,0,0,0,35,0,89,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,0,0,0,0,0,0,91,0,0,0,78,0,0,0,0,22,0,0,1,93,0,0,0,0,19,1,0,0,4,0,0,0,0,70,0,1,0,4,0,1,0,0,86,0,0,0,0,0,0,1,0,8,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,0,34,0,0,0,27] >;
 

C3×C2.F9 in GAP, Magma, Sage, TeX 

C_3\times C_2.F_9
 
% in TeX
 
G:=Group("C3xC2.F9");
 
// GroupNames label
 
G:=SmallGroup(432,565);
 
// by ID
 
G=gap.SmallGroup(432,565);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,3,42,58,80,6053,2371,362,16470,3156,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=1,e^8=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Subgroup lattice of C3×C2.F9 in TeX

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