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## G = C3×C2.F9order 432 = 24·33

### Direct product of C3 and C2.F9

Aliases: C3×C2.F9, C32⋊C48, C6.4F9, C331C16, C2.(C3×F9), (C3×C6).C24, (C32×C6).1C8, C3⋊Dic3.2C12, C322C8.1C6, (C3×C3⋊Dic3).1C4, (C3×C322C8).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 >

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 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 8 8 8 9 9 1 1 8 8 8 9 9 9 9 9 9 9 9 9 ··· 9 9 ··· 9 9 ··· 9

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 8 8 8 8 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 F9 C2.F9 C3×F9 C3×C2.F9 kernel C3×C2.F9 C3×C32⋊2C8 C2.F9 C3×C3⋊Dic3 C32⋊2C8 C32×C6 C3⋊Dic3 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 1 1 2 2

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

 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 0 0 0 1 0 0 0 0 0 0 0 0 0 61 0 0 0 0 0 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 0 0 0 0 0 0 0 0 61 0 0 91 22 14 0 0 11 58 35
,
 1 0 0 0 0 0 0 0 0 0 35 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 35 0 0 0 0 0 0 0 0 0 61 0 0 0 0 0 0 0 0 0 35 0 0 0 0 0 0 0 0 0 1 0 0 6 81 92 27 8 89 0 1
,
 27 0 0 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 91 22 19 70 86 8 51 34 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 78 93 4 4 0 0 0 27

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

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