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## G = C3×C2.PSU3(𝔽2)  order 432 = 24·33

### Direct product of C3 and C2.PSU3(𝔽2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C3×C2.PSU3(𝔽2)
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C6×C3⋊S3 — C6×C32⋊C4 — C3×C2.PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C3×C2.PSU3(𝔽2)
 Upper central C1 — C6

Generators and relations for C3×C2.PSU3(𝔽2)
G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=1, f2=be2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fdf-1=cd=dc, ece-1=d-1, fcf-1=c-1d, ede-1=c, fef-1=be-1 >

Subgroups: 388 in 66 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×C12, C33, C32⋊C4, C32⋊C4, S3×C6, C2×C3⋊S3, C3×C4⋊C4, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C2.PSU3(𝔽2), C6×C32⋊C4, C6×C32⋊C4, C3×C2.PSU3(𝔽2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, C2×C12, C3×D4, C3×Q8, C3×C4⋊C4, PSU3(𝔽2), C2.PSU3(𝔽2), C3×PSU3(𝔽2), C3×C2.PSU3(𝔽2)

Smallest permutation representation of C3×C2.PSU3(𝔽2)
On 48 points
Generators in S48
(1 7 38)(2 8 39)(3 5 40)(4 6 37)(9 14 46)(10 15 47)(11 16 48)(12 13 45)(17 26 42)(18 27 43)(19 28 44)(20 25 41)(21 35 30)(22 36 31)(23 33 32)(24 34 29)
(1 27)(2 28)(3 25)(4 26)(5 41)(6 42)(7 43)(8 44)(9 33)(10 34)(11 35)(12 36)(13 31)(14 32)(15 29)(16 30)(17 37)(18 38)(19 39)(20 40)(21 48)(22 45)(23 46)(24 47)
(2 39 8)(4 6 37)(9 46 14)(10 15 47)(11 16 48)(12 45 13)(17 26 42)(19 44 28)(21 35 30)(22 31 36)(23 32 33)(24 34 29)
(1 7 38)(3 40 5)(9 46 14)(10 47 15)(11 16 48)(12 13 45)(18 27 43)(20 41 25)(21 35 30)(22 36 31)(23 32 33)(24 29 34)
(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)
(1 35 25 9)(2 10 26 36)(3 33 27 11)(4 12 28 34)(5 32 43 16)(6 13 44 29)(7 30 41 14)(8 15 42 31)(17 22 39 47)(18 48 40 23)(19 24 37 45)(20 46 38 21)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A ··· 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A ··· 12L order 1 2 2 2 3 3 3 3 3 4 ··· 4 6 6 6 6 6 6 6 6 6 12 ··· 12 size 1 1 9 9 1 1 8 8 8 18 ··· 18 1 1 8 8 8 9 9 9 9 18 ··· 18

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 8 8 8 8 type + + + - + + image C1 C2 C3 C4 C6 C12 D4 Q8 C3×D4 C3×Q8 PSU3(𝔽2) C2.PSU3(𝔽2) C3×PSU3(𝔽2) C3×C2.PSU3(𝔽2) kernel C3×C2.PSU3(𝔽2) C6×C32⋊C4 C2.PSU3(𝔽2) C3×C32⋊C4 C2×C32⋊C4 C32⋊C4 C3×C3⋊S3 C32×C6 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 3 2 4 6 8 1 1 2 2 1 1 2 2

Matrix representation of C3×C2.PSU3(𝔽2) in GL8(𝔽13)

 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 3 9 0 0 8 12 0 7 11 0 9 0 3 2 4 0 0 0 0 3
,
 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 9 4 3 9 0 0 7 0 3 11 0 0 3 0 0 6 12 5 11 0 0 9
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 3 2 3 11 8 0 0 2 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 5 0 0 0 0 0 12 0 5
,
 0 0 0 0 1 0 0 0 0 0 12 1 1 2 0 0 2 3 3 11 8 0 2 0 3 2 3 11 8 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 10 10 2 0 0 0 0 0 11 10 2

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,8,3,0,1,0,0,0,0,12,2,0,0,9,0,0,0,0,4,0,0,0,3,0,3,7,0,0,0,0,0,3,3,11,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3],[9,0,0,0,0,0,7,0,0,3,0,0,0,0,0,6,0,0,1,0,0,9,3,12,0,0,0,1,0,4,11,5,0,0,0,0,3,3,0,11,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[0,0,0,12,3,0,0,0,0,0,12,0,2,0,0,0,1,0,0,0,3,0,0,0,0,1,0,0,11,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,2,12,5,5],[0,0,2,3,0,0,0,0,0,0,3,2,1,0,0,0,0,12,3,3,0,0,0,0,0,1,11,11,0,0,0,0,1,1,8,8,0,0,0,0,0,2,0,0,0,0,10,11,0,0,2,0,0,1,10,10,0,0,0,2,0,12,2,2] >;

C3×C2.PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_3\times C_2.{\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C3xC2.PSU(3,2)");
// GroupNames label

G:=SmallGroup(432,591);
// by ID

G=gap.SmallGroup(432,591);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,197,260,14117,3036,362,18822,2365,1203]);
// Polycyclic

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

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