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G = C4.2PSU3(𝔽2)  order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C4.2PSU3(𝔽2), C3⋊S3.3D8, C3⋊S3.3Q16, (C3×C12).3Q8, C32⋊2C8⋊2C4, C32⋊1(C2.D8), C2.4(C2.PSU3(𝔽2)), (C3×C6).8(C4⋊C4), (C2×C3⋊S3).11D4, C3⋊S3⋊3C8.6C2, C4⋊(C32⋊C4).5C2, (C4×C3⋊S3).56C22, C3⋊Dic3.15(C2×C4), SmallGroup(288,394)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C4.2PSU3(𝔽2)
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C3⋊S3⋊3C8 — C4.2PSU3(𝔽2)
Lower central
C32 — C3×C6 — C3⋊Dic3 — C4.2PSU3(𝔽2)
Upper central
C1 — C2 — C4

Generators and relations for C4.2PSU3(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=a-1d-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
9C2.D8
C4×C3⋊S3
C32⋊2C8
C32⋊2C8
2C2×C32⋊C4
2C2×C32⋊C4
C3⋊S3⋊3C8
C4⋊(C32⋊C4)
C4⋊(C32⋊C4)
C4.2PSU3(𝔽2)

Character table of C4.2PSU3(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
ρ102-2-222000000-2√2-√2√2-√200    orthogonal lifted from D8
ρ112-2-222000000-2-√2√2-√2√200    orthogonal lifted from D8
ρ122-22-22000000-2-√2√2√2-√200    symplectic lifted from Q16, Schur index 2
ρ132-22-22000000-2√2-√2-√2√200    symplectic lifted from Q16, Schur index 2
ρ1422-2-222-200002000022    symplectic lifted from Q8, Schur index 2
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 C4.2PSU3(𝔽2)
►On 48 points
Generators in S48
(1 7 5 3)(2 8 6 4)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)(33 39 37 35)(34 40 38 36)

(2 24 10)(4 12 18)(6 20 14)(8 16 22)(25 44 34)(26 45 35)(27 36 46)(28 37 47)(29 48 38)(30 41 39)(31 40 42)(32 33 43)

(1 23 9)(2 24 10)(3 11 17)(4 12 18)(5 19 13)(6 20 14)(7 15 21)(8 16 22)(25 34 44)(27 46 36)(29 38 48)(31 42 40)

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

(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)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(33,39,37,35)(34,40,38,36), (2,24,10)(4,12,18)(6,20,14)(8,16,22)(25,44,34)(26,45,35)(27,36,46)(28,37,47)(29,48,38)(30,41,39)(31,40,42)(32,33,43), (1,23,9)(2,24,10)(3,11,17)(4,12,18)(5,19,13)(6,20,14)(7,15,21)(8,16,22)(25,34,44)(27,46,36)(29,38,48)(31,42,40), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,30,23,41)(10,29,24,48)(11,28,17,47)(12,27,18,46)(13,26,19,45)(14,25,20,44)(15,32,21,43)(16,31,22,42), (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)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(33,39,37,35)(34,40,38,36), (2,24,10)(4,12,18)(6,20,14)(8,16,22)(25,44,34)(26,45,35)(27,36,46)(28,37,47)(29,48,38)(30,41,39)(31,40,42)(32,33,43), (1,23,9)(2,24,10)(3,11,17)(4,12,18)(5,19,13)(6,20,14)(7,15,21)(8,16,22)(25,34,44)(27,46,36)(29,38,48)(31,42,40), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,40)(9,30,23,41)(10,29,24,48)(11,28,17,47)(12,27,18,46)(13,26,19,45)(14,25,20,44)(15,32,21,43)(16,31,22,42), (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),(9,21,13,17),(10,22,14,18),(11,23,15,19),(12,24,16,20),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41),(33,39,37,35),(34,40,38,36)], [(2,24,10),(4,12,18),(6,20,14),(8,16,22),(25,44,34),(26,45,35),(27,36,46),(28,37,47),(29,48,38),(30,41,39),(31,40,42),(32,33,43)], [(1,23,9),(2,24,10),(3,11,17),(4,12,18),(5,19,13),(6,20,14),(7,15,21),(8,16,22),(25,34,44),(27,46,36),(29,38,48),(31,42,40)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,40),(9,30,23,41),(10,29,24,48),(11,28,17,47),(12,27,18,46),(13,26,19,45),(14,25,20,44),(15,32,21,43),(16,31,22,42)], [(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 C4.2PSU3(𝔽2) ►in GL10(𝔽73)

72200000000
72100000000
00720000000
00072000000
00007200000
00000720000
00000072000
00000007200
00000000720
00000000072
,
1000000000
0100000000
0010000000
0001000000
00000720000
00001720000
00000590100
0077140727200
005959660007272
0000070010
,
1000000000
0100000000
00072000000
00172000000
0000100000
0000010000
00066000100
00701414727200
00014000001
005906666007272
,
465400000000
02700000000
0000100000
0000010000
0001000000
0010000000
00000000720
001414770011
00000072000
00000007200
,
04100000000
164100000000
00000072100
00771414717200
00000000721
0059596666007172
000000660590
0007200660590
00000014070
000007214070

G:=sub<GL(10,GF(73))| [72,72,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7,59,0,0,0,0,1,0,0,0,7,59,0,0,0,0,0,0,1,0,14,66,0,0,0,0,0,72,72,59,0,0,7,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,7,0,59,0,0,72,72,0,0,66,0,14,0,0,0,0,0,1,0,0,14,0,66,0,0,0,0,0,1,0,14,0,66,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[46,0,0,0,0,0,0,0,0,0,54,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,14,0,0,0,0,0,0,1,0,0,14,0,0,0,0,1,0,0,0,0,7,0,0,0,0,0,1,0,0,0,7,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,0,0,41,41,0,0,0,0,0,0,0,0,0,0,0,7,0,59,0,0,0,0,0,0,0,7,0,59,0,72,0,0,0,0,0,14,0,66,0,0,0,0,0,0,0,14,0,66,0,0,0,72,0,0,72,71,0,0,66,66,14,14,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,71,59,59,7,7,0,0,0,0,1,72,0,0,0,0] >;
 

C4.2PSU3(𝔽2) in GAP, Magma, Sage, TeX 

C_4._2{\rm PSU}_3({\mathbb F}_2)
 
% in TeX
 
G:=Group("C4.2PSU(3,2)");
 
// GroupNames label
 
G:=SmallGroup(288,394);
 
// by ID
 
G=gap.SmallGroup(288,394);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,85,92,219,100,346,80,9413,2028,691,12550,1581,2372]);
 
// 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^-1*d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.2PSU3(𝔽2) in TeX
Character table of C4.2PSU3(𝔽2) in TeX

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