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G = C6.2PSU3(𝔽2)  order 432 = 24·33

2nd non-split extension by C6 of PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2

non-abelian, soluble, monomial

Aliases: C6.2PSU3(𝔽2), C337(C4⋊C4), C3⋊S3.3D12, C32⋊C42Dic3, (C3×C6).4Dic6, (C32×C6).3Q8, C323(C4⋊Dic3), C2.2(C33⋊Q8), C32(C2.PSU3(𝔽2)), (C3×C32⋊C4)⋊2C4, (C3×C3⋊S3).13D4, (C2×C3⋊S3).16D6, (C6×C32⋊C4).6C2, (C2×C32⋊C4).4S3, C3⋊S3.4(C2×Dic3), (C6×C3⋊S3).13C22, (C2×C33⋊C4).5C2, (C3×C3⋊S3).14(C2×C4), SmallGroup(432,593)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C6.2PSU3(𝔽2)
C1C3C33C3×C3⋊S3C6×C3⋊S3C2×C33⋊C4 — C6.2PSU3(𝔽2)
C33C3×C3⋊S3 — C6.2PSU3(𝔽2)
C1C2

Generators and relations for C6.2PSU3(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=d-1 >

Subgroups: 528 in 66 conjugacy classes, 23 normal (17 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, C32⋊C4, S3×C6, C2×C3⋊S3, C4⋊Dic3, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C2.PSU3(𝔽2), C6×C32⋊C4, C2×C33⋊C4, C6.2PSU3(𝔽2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, C4⋊Dic3, PSU3(𝔽2), C2.PSU3(𝔽2), C33⋊Q8, C6.2PSU3(𝔽2)

Character table of C6.2PSU3(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
ρ51-11-111111-1-ii-ii-1-1-1-1-11-1-111    linear of order 4
ρ61-11-111111-1i-ii-i-1-1-1-1-11-1-111    linear of order 4
ρ71-11-11111-11-i-iii-1-1-1-1-1111-1-1    linear of order 4
ρ81-11-11111-11ii-i-i-1-1-1-1-1111-1-1    linear of order 4
ρ92222-1-12-1220000-1-1-12-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-2-222222000000-2-2-2-22-20000    orthogonal lifted from D4
ρ112-2-22-1-12-1000000111-2-113-33-3    orthogonal lifted from D12
ρ122222-1-12-1-2-20000-1-1-12-1-11111    orthogonal lifted from D6
ρ132-2-22-1-12-1000000111-2-11-33-33    orthogonal lifted from D12
ρ142-22-2-1-12-1-220000111-21-1-1-111    symplectic lifted from Dic3, Schur index 2
ρ1522-2-2-1-12-1000000-1-1-1211-333-3    symplectic lifted from Dic6, Schur index 2
ρ162-22-2-1-12-12-20000111-21-111-1-1    symplectic lifted from Dic3, Schur index 2
ρ1722-2-2-1-12-1000000-1-1-12113-3-33    symplectic lifted from Dic6, Schur index 2
ρ1822-2-222220000002222-2-20000    symplectic lifted from Q8, Schur index 2
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 C6.2PSU3(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 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)
(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 37 34 46)(26 38 35 47)(27 39 36 48)(28 40 31 43)(29 41 32 44)(30 42 33 45)
(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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34), (1,22,10,13)(2,23,11,14)(3,24,12,15)(4,19,7,16)(5,20,8,17)(6,21,9,18)(25,37,34,46)(26,38,35,47)(27,39,36,48)(28,40,31,43)(29,41,32,44)(30,42,33,45), (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)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,35,33),(32,36,34)], [(1,22,10,13),(2,23,11,14),(3,24,12,15),(4,19,7,16),(5,20,8,17),(6,21,9,18),(25,37,34,46),(26,38,35,47),(27,39,36,48),(28,40,31,43),(29,41,32,44),(30,42,33,45)], [(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 C6.2PSU3(𝔽2) in GL10(𝔽13)

11200000000
1000000000
0090000000
0009000000
0000900000
0000090000
0000203000
00001100300
0020000030
00110000003
,
1000000000
0100000000
0090000000
0033000000
0000900000
0000330000
0000701000
0000600100
0020000030
0000000009
,
1000000000
0100000000
0090000000
0033000000
0000300000
00001090000
00001109000
0000000300
0070000010
0060000001
,
10600000000
7300000000
0000100000
0000010000
00122000000
0001000000
0005000001
0008000010
0000001000
0000000100
,
7300000000
10600000000
0000802000
0000001100
0080000020
0000000011
0001000050
0000000080
0000015000
0000008000

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

C6.2PSU3(𝔽2) in GAP, Magma, Sage, TeX

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

G:=Group("C6.2PSU(3,2)");
// GroupNames label

G:=SmallGroup(432,593);
// by ID

G=gap.SmallGroup(432,593);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,197,92,2804,1691,298,2693,348,1027,14118]);
// 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=d^-1>;
// generators/relations

Export

Character table of C6.2PSU3(𝔽2) in TeX

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