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G = C6.PSU3(𝔽2)  order 432 = 24·33

1st non-split extension by C6 of PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2

non-abelian, soluble, monomial

Aliases: C6.1PSU3(𝔽2), C33⋊6(C4⋊C4), C33⋊C4⋊3C4, (C3×C6).3Dic6, (C32×C6).2Q8, C2.1(C33⋊Q8), C32⋊4(Dic3⋊C4), C3⋊1(C2.PSU3(𝔽2)), C3⋊S3.5(C4×S3), (C2×C3⋊S3).15D6, (C3×C3⋊S3).12D4, (C6×C32⋊C4).2C2, (C2×C32⋊C4).3S3, C3⋊S3.5(C3⋊D4), (C6×C3⋊S3).12C22, (C2×C33⋊C4).4C2, (C3×C3⋊S3).13(C2×C4), SmallGroup(432,592)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3 — C6.PSU3(𝔽2)
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C33⋊C4 — C6.PSU3(𝔽2)
Lower central
C33 — C3×C3⋊S3 — C6.PSU3(𝔽2)
Upper central
C1 — C2

Generators and relations for C6.PSU3(𝔽2)
 G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=a3d2, ab=ba, ac=ca, ad=da, eae-1=a-1, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=a3d-1 >

Subgroups: 528 in 66 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C33, C32⋊C4, S3×C6, C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C33⋊C4, C6×C3⋊S3, C2.PSU3(𝔽2), C6×C32⋊C4, C2×C33⋊C4, C6.PSU3(𝔽2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, Dic3⋊C4, PSU3(𝔽2), C2.PSU3(𝔽2), C33⋊Q8, C6.PSU3(𝔽2)

Character table of C6.PSU3(𝔽2)

 class 12A2B2C3A3B3C3D4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D
 size 119928881818545454542888181818181818
ρ1111111111111111111111111    trivial
ρ21111111111-1-1-1-11111111111    linear of order 2
ρ311111111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ411111111-1-1-111-1111111-1-1-1-1    linear of order 2
ρ51-1-111111-iii1-1-i-1-1-1-11-1ii-i-i    linear of order 4
ρ61-1-111111-ii-i-11i-1-1-1-11-1ii-i-i    linear of order 4
ρ71-1-111111i-ii-11-i-1-1-1-11-1-i-iii    linear of order 4
ρ81-1-111111i-i-i1-1i-1-1-1-11-1-i-iii    linear of order 4
ρ92222-1-12-1220000-1-1-12-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-22-22222000000-2-2-2-2-220000    orthogonal lifted from D4
ρ112222-1-12-1-2-20000-1-1-12-1-11111    orthogonal lifted from D6
ρ1222-2-222220000002222-2-20000    symplectic lifted from Q8, Schur index 2
ρ1322-2-2-1-12-1000000-1-1-1211-√3√3√3-√3    symplectic lifted from Dic6, Schur index 2
ρ1422-2-2-1-12-1000000-1-1-1211√3-√3-√3√3    symplectic lifted from Dic6, Schur index 2
ρ152-2-22-1-12-12i-2i0000111-2-11ii-i-i    complex lifted from C4×S3
ρ162-2-22-1-12-1-2i2i0000111-2-11-i-iii    complex lifted from C4×S3
ρ172-22-2-1-12-1000000111-21-1√-3-√-3√-3-√-3    complex lifted from C3⋊D4
ρ182-22-2-1-12-1000000111-21-1-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ198-8008-1-1-1000000-8111000000    orthogonal lifted from C2.PSU3(𝔽2)
ρ2088008-1-1-10000008-1-1-1000000    orthogonal lifted from PSU3(𝔽2)
ρ218-800-41-3√-3/2-11+3√-3/20000004-1-3√-3/2-1+3√-3/21000000    complex faithful
ρ228800-41-3√-3/2-11+3√-3/2000000-41+3√-3/21-3√-3/2-1000000    complex lifted from C33⋊Q8
ρ238-800-41+3√-3/2-11-3√-3/20000004-1+3√-3/2-1-3√-3/21000000    complex faithful
ρ248800-41+3√-3/2-11-3√-3/2000000-41-3√-3/21+3√-3/2-1000000    complex lifted from C33⋊Q8

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)

(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(37 39 41)(38 40 42)(43 47 45)(44 48 46)

(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 33 35)(32 34 36)

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

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

Matrix representation of C6.PSU3(𝔽2) ►in GL8(𝔽13)

100000000
010000000
001000000
000100000
00004000
99000400
610400040
26400004
,
90000000
03000000
00900000
40930000
00001000
123000100
604010030
070012009
,
90000000
03000000
00300000
04490000
00009000
90004300
81210012010
721001001
,
00100000
121180000
012000000
000120000
6512012008
816120011
3212900012
1135801201
,
00001000
110012800
541201080
0000012121
01000000
000001200
00010910
00000810

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

C6.PSU3(𝔽2) in GAP, Magma, Sage, TeX 

C_6.{\rm PSU}_3({\mathbb F}_2)
 
% in TeX
 
G:=Group("C6.PSU(3,2)");
 
// GroupNames label
 
G:=SmallGroup(432,592);
 
// by ID
 
G=gap.SmallGroup(432,592);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,85,92,2804,1691,298,2693,348,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=a^3*d^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,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=a^3*d^-1>;
 
// generators/relations
 

Export

Character table of C6.PSU3(𝔽2) in TeX

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