Copied to
clipboard

G = C3×C2.PSU3(𝔽2)  order 432 = 24·33

Direct product of C3 and C2.PSU3(𝔽2)

direct product, non-abelian, soluble, monomial

Aliases: C3×C2.PSU3(𝔽2), C6.5PSU3(𝔽2), C32⋊C4⋊2C12, C33⋊5(C4⋊C4), (C32×C6).1Q8, C2.(C3×PSU3(𝔽2)), (C3×C6).(C3×Q8), C32⋊2(C3×C4⋊C4), (C3×C32⋊C4)⋊5C4, (C3×C3⋊S3).6D4, C3⋊S3.3(C3×D4), C3⋊S3.4(C2×C12), (C2×C32⋊C4).2C6, (C6×C32⋊C4).11C2, (C6×C3⋊S3).11C22, (C3×C3⋊S3).6(C2×C4), (C2×C3⋊S3).6(C2×C6), SmallGroup(432,591)

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 2A2B2C3A3B3C3D3E4A···4F6A6B6C6D6E6F6G6H6I12A···12L
order1222333334···466666666612···12
size11991188818···1811888999918···18

36 irreducible representations

dim11111122228888
type+++-++
imageC1C2C3C4C6C12D4Q8C3×D4C3×Q8PSU3(𝔽2)C2.PSU3(𝔽2)C3×PSU3(𝔽2)C3×C2.PSU3(𝔽2)
kernelC3×C2.PSU3(𝔽2)C6×C32⋊C4C2.PSU3(𝔽2)C3×C32⋊C4C2×C32⋊C4C32⋊C4C3×C3⋊S3C32×C6C3⋊S3C3×C6C6C3C2C1
# reps13246811221122

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

90000000
09000000
00900000
00090000
00009000
00000900
00000090
00000009
,
120000000
012000000
001200000
000120000
000012000
000001200
000000120
000000012
,
10000000
01000000
00900000
00030000
00003000
00033900
8120711090
32400003
,
90000000
03000000
00100000
00010000
00003000
00943900
703110030
0612511009
,
00100000
00010000
012000000
120000000
323118002
000000112
00000005
000001205
,
00001000
001211200
233118020
323118002
01000000
000000112
0000010102
0000011102

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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁