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G = C4.4PSU3(𝔽2)  order 288 = 25·32

The central extension by C4 of PSU3(𝔽2)

non-abelian, soluble, monomial

Aliases: C4.4PSU3(𝔽2), C32⋊C4⋊3C8, C32⋊3(C4⋊C8), (C3×C12).1Q8, C3⋊Dic3.7D4, C3⋊S3.4M4(2), C2.1(C2.PSU3(𝔽2)), C3⋊S3.4(C2×C8), (C3×C6).6(C4⋊C4), (C4×C32⋊C4).4C2, (C2×C32⋊C4).2C4, C3⋊S3⋊3C8.4C2, (C4×C3⋊S3).54C22, (C2×C3⋊S3).11(C2×C4), SmallGroup(288,392)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊S3 — C4.4PSU3(𝔽2)
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4×C32⋊C4 — C4.4PSU3(𝔽2)
Lower central
C32 — C3⋊S3 — C4.4PSU3(𝔽2)
Upper central
C1 — C4

Generators and relations for C4.4PSU3(𝔽2)
 G = < a,b,c,d,e | a4=b3=c3=1, d4=a2, e2=a-1d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=a-1d3 >

C1
C2
9C2
9C2
4C3
C4
9C4
9C4
9C22
9C4
18C4
4C6
12S3
12S3
C32
9C2×C4
9C2×C4
9C2×C4
18C8
18C8
4C12
12D6
12Dic3
C3⋊S3
C3⋊S3
C3×C6
9C2×C8
9C2×C8
9C42
12C4×S3
C3⋊Dic3
C32⋊C4
C2×C3⋊S3
C32⋊C4
C3×C12
2C32⋊C4
9C4⋊C8
C4×C3⋊S3
C2×C32⋊C4
C2×C32⋊C4
2C32⋊2C8
2C32⋊2C8
C3⋊S3⋊3C8
C4×C32⋊C4
C3⋊S3⋊3C8
C4.4PSU3(𝔽2)

Character table of C4.4PSU3(𝔽2)

 class 12A2B2C34A4B4C4D4E4F4G4H68A8B8C8D8E8F8G8H12A12B
 size 119981199181818188181818181818181888
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-111-1-1-1111-111    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ4111111111-1-1-1-11-1111-1-1-1111    linear of order 2
ρ511111-1-1-1-111-1-11i-iii-i-ii-i-1-1    linear of order 4
ρ611111-1-1-1-111-1-11-ii-i-iii-ii-1-1    linear of order 4
ρ711111-1-1-1-1-1-1111ii-i-i-i-iii-1-1    linear of order 4
ρ811111-1-1-1-1-1-1111-i-iiiii-i-i-1-1    linear of order 4
ρ91-1-111-ii-ii-11-ii-1ζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85i-i    linear of order 8
ρ101-1-111-ii-ii-11-ii-1ζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8i-i    linear of order 8
ρ111-1-111i-ii-i1-1-ii-1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83-ii    linear of order 8
ρ121-1-111i-ii-i1-1-ii-1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87-ii    linear of order 8
ρ131-1-111i-ii-i-11i-i-1ζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87-ii    linear of order 8
ρ141-1-111i-ii-i-11i-i-1ζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83-ii    linear of order 8
ρ151-1-111-ii-ii1-1i-i-1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85i-i    linear of order 8
ρ161-1-111-ii-ii1-1i-i-1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8i-i    linear of order 8
ρ1722-2-22-2-2220000200000000-2-2    orthogonal lifted from D4
ρ1822-2-2222-2-2000020000000022    symplectic lifted from Q8, Schur index 2
ρ192-22-22-2i2i2i-2i0000-2000000002i-2i    complex lifted from M4(2)
ρ202-22-222i-2i-2i2i0000-200000000-2i2i    complex lifted from M4(2)
ρ218800-1-8-8000000-10000000011    orthogonal lifted from C2.PSU3(𝔽2)
ρ228800-188000000-100000000-1-1    orthogonal lifted from PSU3(𝔽2)
ρ238-800-18i-8i000000100000000i-i    complex faithful
ρ248-800-1-8i8i000000100000000-ii    complex faithful

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

(2 35 25)(4 27 37)(6 39 29)(8 31 33)(9 48 20)(10 41 21)(11 22 42)(12 23 43)(13 44 24)(14 45 17)(15 18 46)(16 19 47)

(1 32 34)(3 36 26)(5 28 38)(7 40 30)(9 20 48)(10 41 21)(11 42 22)(12 23 43)(13 24 44)(14 45 17)(15 46 18)(16 19 47)

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

Matrix representation of C4.4PSU3(𝔽2) ►in GL10(𝔽73)

46000000000
04600000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
00007210000
00007200000
00000007200
00000017200
000072004601
002525012707272
,
1000000000
0100000000
00721000000
00720000000
0000100000
0000010000
00000007200
00000017200
00025112707272
004800004610
,
10000000000
06300000000
0000100000
0000010000
0001000000
0010000000
0025251127277172
00000000721
00555566664848460
00555566664948460
,
07200000000
1000000000
0000001000
0000000100
00000000721
0025251127277172
0001000000
0010000000
006666006666720
006666016666720

G:=sub<GL(10,GF(73))| [46,0,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,25,0,0,0,1,0,0,0,0,0,25,0,0,0,0,72,72,0,0,72,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,27,0,0,0,0,0,0,72,72,46,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,48,0,0,1,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,27,0,0,0,0,0,0,0,72,72,0,46,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[10,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,0,0,0,1,25,0,55,55,0,0,0,0,1,0,25,0,55,55,0,0,1,0,0,0,1,0,66,66,0,0,0,1,0,0,1,0,66,66,0,0,0,0,0,0,27,0,48,49,0,0,0,0,0,0,27,0,48,48,0,0,0,0,0,0,71,72,46,46,0,0,0,0,0,0,72,1,0,0],[0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,1,66,66,0,0,0,0,0,25,1,0,66,66,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,27,0,0,66,66,0,0,0,1,0,27,0,0,66,66,0,0,0,0,72,71,0,0,72,72,0,0,0,0,1,72,0,0,0,0] >;
 

C4.4PSU3(𝔽2) in GAP, Magma, Sage, TeX 

C_4._4{\rm PSU}_3({\mathbb F}_2)
 
% in TeX
 
G:=Group("C4.4PSU(3,2)");
 
// GroupNames label
 
G:=SmallGroup(288,392);
 
// by ID
 
G=gap.SmallGroup(288,392);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,85,92,80,9413,2028,691,12550,1581,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=1,d^4=a^2,e^2=a^-1*d^2,a*b=b*a,a*c=c*a,a*d=d*a,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^3>;
 
// generators/relations
 

Export

Subgroup lattice of C4.4PSU3(𝔽2) in TeX
Character table of C4.4PSU3(𝔽2) in TeX

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