Copied to
clipboard

G = C4.PSU3(𝔽2)  order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C4.1PSU3(𝔽2), (C3×C12).2Q8, C32⋊2C8⋊4C4, C3⋊S3.4SD16, C32⋊2(C4.Q8), C2.3(C2.PSU3(𝔽2)), (C3×C6).7(C4⋊C4), (C2×C3⋊S3).10D4, C3⋊S3⋊3C8.5C2, C4⋊(C32⋊C4).4C2, (C4×C3⋊S3).55C22, C3⋊Dic3.14(C2×C4), SmallGroup(288,393)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C4.PSU3(𝔽2)
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C3⋊S3⋊3C8 — C4.PSU3(𝔽2)
Lower central
C32 — C3×C6 — C3⋊Dic3 — C4.PSU3(𝔽2)
Upper central
C1 — C2 — C4

Generators and relations for C4.PSU3(𝔽2)
 G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=a-1d2, ab=ba, ac=ca, dad-1=a-1, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=ad-1 >

C1
C2
9C2
9C2
4C3
C4
9C4
9C22
36C4
36C4
4C6
12S3
12S3
C32
9C8
9C8
9C2×C4
18C2×C4
18C2×C4
4C12
12D6
12Dic3
C3⋊S3
C3×C6
C3⋊S3
9C2×C8
9C4⋊C4
9C4⋊C4
12C4×S3
C3×C12
C3⋊Dic3
C2×C3⋊S3
4C32⋊C4
4C32⋊C4
9C4.Q8
C4×C3⋊S3
C32⋊2C8
C32⋊2C8
2C2×C32⋊C4
2C2×C32⋊C4
C3⋊S3⋊3C8
C4⋊(C32⋊C4)
C4⋊(C32⋊C4)
C4.PSU3(𝔽2)

Character table of C4.PSU3(𝔽2)

 class 12A2B2C34A4B4C4D4E4F68A8B8C8D12A12B
 size 119982183636363681818181888
ρ1111111111111111111    trivial
ρ21111111-11-111-1-1-1-111    linear of order 2
ρ31111111-1-1-1-11111111    linear of order 2
ρ411111111-11-11-1-1-1-111    linear of order 2
ρ511-1-11-11-i-iii111-1-1-1-1    linear of order 4
ρ611-1-11-11-iii-i1-1-111-1-1    linear of order 4
ρ711-1-11-11i-i-ii1-1-111-1-1    linear of order 4
ρ811-1-11-11ii-i-i111-1-1-1-1    linear of order 4
ρ922222-2-2000020000-2-2    orthogonal lifted from D4
ρ1022-2-222-200002000022    symplectic lifted from Q8, Schur index 2
ρ112-22-22000000-2√-2-√-2√-2-√-200    complex lifted from SD16
ρ122-2-222000000-2√-2-√-2-√-2√-200    complex lifted from SD16
ρ132-2-222000000-2-√-2√-2√-2-√-200    complex lifted from SD16
ρ142-22-22000000-2-√-2√-2-√-2√-200    complex lifted from SD16
ρ158800-1-800000-1000011    orthogonal lifted from C2.PSU3(𝔽2)
ρ168800-1800000-10000-1-1    orthogonal lifted from PSU3(𝔽2)
ρ178-800-100000010000-3i3i    complex faithful
ρ188-800-1000000100003i-3i    complex faithful

Smallest permutation representation of C4.PSU3(𝔽2)
►On 48 points
Generators in S48
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)

(2 20 34)(4 36 22)(6 24 38)(8 40 18)(9 43 32)(10 25 44)(11 26 45)(12 46 27)(13 47 28)(14 29 48)(15 30 41)(16 42 31)

(1 19 33)(2 20 34)(3 35 21)(4 36 22)(5 23 37)(6 24 38)(7 39 17)(8 40 18)(10 44 25)(12 27 46)(14 48 29)(16 31 42)

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

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

Matrix representation of C4.PSU3(𝔽2) ►in GL10(𝔽73)

72200000000
72100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
00000720000
00001720000
00000072100
00000072000
00000000072
00000000172
,
1000000000
0100000000
00072000000
00172000000
0000100000
0000010000
00000072100
00000072000
00000000721
00000000720
,
713500000000
52200000000
0000100000
0000010000
0001000000
0010000000
0000000001
0000000010
0000001000
0000000100
,
611200000000
67000000000
0000001000
0000000100
0000000010
0000000001
0001000000
0010000000
0000010000
0000100000

G:=sub<GL(10,GF(73))| [72,72,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[71,52,0,0,0,0,0,0,0,0,35,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[61,67,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C4.PSU3(𝔽2) in GAP, Magma, Sage, TeX 

C_4.{\rm PSU}_3({\mathbb F}_2)
 
% in TeX
 
G:=Group("C4.PSU(3,2)");
 
// GroupNames label
 
G:=SmallGroup(288,393);
 
// by ID
 
G=gap.SmallGroup(288,393);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,219,100,346,80,9413,2028,691,12550,1581,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=1,e^2=a^-1*d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,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=a*d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.PSU3(𝔽2) in TeX
Character table of C4.PSU3(𝔽2) in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁