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G = C6×PSU3(𝔽2)  order 432 = 24·33

Direct product of C6 and PSU3(𝔽2)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×PSU3(𝔽2)
G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >

Subgroups: 468 in 90 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×C12, C3×Q8, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C6×Q8, C3×C3⋊S3, C32×C6, PSU3(𝔽2), C2×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C2×PSU3(𝔽2), C3×PSU3(𝔽2), C6×C32⋊C4, C6×PSU3(𝔽2)
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C3×Q8, C22×C6, C6×Q8, PSU3(𝔽2), C2×PSU3(𝔽2), C3×PSU3(𝔽2), C6×PSU3(𝔽2)

Smallest permutation representation of C6×PSU3(𝔽2)
On 48 points
Generators in S48
(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)
(7 11 9)(8 12 10)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 31 43 39)(8 32 44 40)(9 33 45 41)(10 34 46 42)(11 35 47 37)(12 36 48 38)
(1 37 16 35)(2 38 17 36)(3 39 18 31)(4 40 13 32)(5 41 14 33)(6 42 15 34)(7 30 43 22)(8 25 44 23)(9 26 45 24)(10 27 46 19)(11 28 47 20)(12 29 48 21)

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

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

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

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 C2 C3 C6 C6 Q8 Q8 C3×Q8 C3×Q8 PSU3(𝔽2) C2×PSU3(𝔽2) C3×PSU3(𝔽2) C6×PSU3(𝔽2) kernel C6×PSU3(𝔽2) C3×PSU3(𝔽2) C6×C32⋊C4 C2×PSU3(𝔽2) PSU3(𝔽2) C2×C32⋊C4 C3×C3⋊S3 C32×C6 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 4 3 2 8 6 1 1 2 2 1 1 2 2

Matrix representation of C6×PSU3(𝔽2) in GL10(𝔽13)

 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 9 0 0 0 0 0 0 0 12 0 0 3 0 0 0 0 0 0 1 0 0 0 3 0 0 0 0 0 4 0 0 0 0 9 0 0 0 0 9 0 0 0 0 0 9 0 0 0 12 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 0 10 0 0 0 3 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 9
,
 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 2 0 0 0 0 0 0 0 0 0 12 1 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 12 0 0 0 0 1 0 0 0 0 12 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0
,
 3 9 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0

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

C6×PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_6\times {\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C6xPSU(3,2)");
// GroupNames label

G:=SmallGroup(432,757);
// by ID

G=gap.SmallGroup(432,757);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,3,168,365,176,14117,3036,201,18822,2365,622]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations

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