Copied to
clipboard

G = C2×C2.PSU3(𝔽2)  order 288 = 25·32

Direct product of C2 and C2.PSU3(𝔽2)

direct product, non-abelian, soluble, monomial

Aliases: C2×C2.PSU3(𝔽2), C62.3Q8, C22.3PSU3(𝔽2), C2.2(C2×PSU3(𝔽2)), C322(C2×C4⋊C4), C3⋊S32(C4⋊C4), (C3×C6)⋊2(C4⋊C4), (C2×C32⋊C4)⋊5C4, C32⋊C43(C2×C4), C3⋊S3.8(C2×D4), (C2×C3⋊S3).4Q8, (C3×C6).4(C2×Q8), (C2×C3⋊S3).37D4, C3⋊S3.6(C22×C4), (C2×C3⋊S3).15C23, (C22×C32⋊C4).6C2, (C2×C32⋊C4).20C22, (C22×C3⋊S3).54C22, (C2×C3⋊S3).19(C2×C4), SmallGroup(288,894)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C2×C2.PSU3(𝔽2)
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C2.PSU3(𝔽2) — C2×C2.PSU3(𝔽2)
C32C3⋊S3 — C2×C2.PSU3(𝔽2)
C1C22

Generators and relations for C2×C2.PSU3(𝔽2)
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=1, f2=be2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fdf-1=cd=dc, ece-1=d-1, fcf-1=c-1d, ede-1=c, fef-1=be-1 >

Subgroups: 652 in 108 conjugacy classes, 43 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×8], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4 [×14], C23, C32, D6 [×6], C2×C6, C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C22×S3, C2×C4⋊C4, C32⋊C4 [×4], C32⋊C4 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62, C2×C32⋊C4 [×10], C2×C32⋊C4 [×4], C22×C3⋊S3, C2.PSU3(𝔽2) [×4], C22×C32⋊C4, C22×C32⋊C4 [×2], C2×C2.PSU3(𝔽2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, PSU3(𝔽2), C2.PSU3(𝔽2) [×2], C2×PSU3(𝔽2), C2×C2.PSU3(𝔽2)

Character table of C2×C2.PSU3(𝔽2)

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C
 size 111199998181818181818181818181818888
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-11111-1-1-1-1111    linear of order 2
ρ311-1-1-11-11111-1-1-1-111-1-111-1-11    linear of order 2
ρ411-1-1-11-111-1-111-1-11111-1-1-1-11    linear of order 2
ρ51111111111-1-1-1-1-1-1-1111-1111    linear of order 2
ρ6111111111-1111-1-1-1-1-1-1-11111    linear of order 2
ρ711-1-1-11-1111-11111-1-1-1-11-1-1-11    linear of order 2
ρ811-1-1-11-111-11-1-111-1-111-11-1-11    linear of order 2
ρ91-11-1-1-1111i-ii-i-11-11-ii-ii-11-1    linear of order 4
ρ101-11-1-1-1111-ii-ii-11-11i-ii-i-11-1    linear of order 4
ρ111-1-111-1-111i-i-ii1-1-11i-i-ii1-1-1    linear of order 4
ρ121-1-111-1-111-iii-i1-1-11-iii-i1-1-1    linear of order 4
ρ131-11-1-1-1111ii-ii1-11-1-ii-i-i-11-1    linear of order 4
ρ141-11-1-1-1111-i-ii-i1-11-1i-iii-11-1    linear of order 4
ρ151-1-111-1-111iii-i-111-1i-i-i-i1-1-1    linear of order 4
ρ161-1-111-1-111-i-i-ii-111-1-iiii1-1-1    linear of order 4
ρ172-22-222-2-22000000000000-22-2    orthogonal lifted from D4
ρ182-2-22-222-220000000000002-2-2    orthogonal lifted from D4
ρ192222-2-2-2-22000000000000222    symplectic lifted from Q8, Schur index 2
ρ2022-2-22-22-22000000000000-2-22    symplectic lifted from Q8, Schur index 2
ρ218-8-880000-1000000000000-111    orthogonal lifted from C2.PSU3(𝔽2)
ρ2288-8-80000-100000000000011-1    orthogonal lifted from C2×PSU3(𝔽2)
ρ2388880000-1000000000000-1-1-1    orthogonal lifted from PSU3(𝔽2)
ρ248-88-80000-10000000000001-11    orthogonal lifted from C2.PSU3(𝔽2)

Smallest permutation representation of C2×C2.PSU3(𝔽2)
On 48 points
Generators in S48
(1 12)(2 11)(3 6)(4 5)(7 13)(8 14)(9 15)(10 16)(17 40)(18 37)(19 38)(20 39)(21 43)(22 44)(23 41)(24 42)(25 32)(26 29)(27 30)(28 31)(33 46)(34 47)(35 48)(36 45)
(1 5)(2 6)(3 11)(4 12)(7 9)(8 10)(13 15)(14 16)(17 42)(18 43)(19 44)(20 41)(21 37)(22 38)(23 39)(24 40)(25 45)(26 46)(27 47)(28 48)(29 33)(30 34)(31 35)(32 36)
(2 40 38)(3 42 44)(6 24 22)(7 33 35)(8 34 36)(9 29 31)(10 30 32)(11 17 19)(13 46 48)(14 47 45)(15 26 28)(16 27 25)
(1 37 39)(4 43 41)(5 21 23)(7 35 33)(8 34 36)(9 31 29)(10 30 32)(12 18 20)(13 48 46)(14 47 45)(15 28 26)(16 27 25)
(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 14 5 16)(2 15 6 13)(3 7 11 9)(4 10 12 8)(17 31 44 33)(18 34 41 32)(19 29 42 35)(20 36 43 30)(21 27 39 45)(22 46 40 28)(23 25 37 47)(24 48 38 26)

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

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

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

Matrix representation of C2×C2.PSU3(𝔽2) in GL10(𝔽13)

12000000000
01200000000
00120000000
00012000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
1000000000
0100000000
00120000000
00012000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
1000000000
0100000000
0010000000
0001000000
00000120000
00001120000
00000012100
00000012000
00000000012
00000000112
,
1000000000
0100000000
00012000000
00112000000
0000100000
0000010000
00000012100
00000012000
00000000121
00000000120
,
5000000000
0800000000
00001200000
00000120000
00012000000
00120000000
0000000001
0000000010
0000001000
0000000100
,
0100000000
12000000000
0000001000
0000000100
0000000010
0000000001
00012000000
00120000000
00000120000
00001200000

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0],[5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C2×C2.PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_2\times C_2.{\rm PSU}_3({\mathbb F}_2)
% in TeX

G:=Group("C2xC2.PSU(3,2)");
// GroupNames label

G:=SmallGroup(288,894);
// by ID

G=gap.SmallGroup(288,894);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,141,176,9413,2028,362,12550,1581,1203]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^4=1,f^2=b*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*d*f^-1=c*d=d*c,e*c*e^-1=d^-1,f*c*f^-1=c^-1*d,e*d*e^-1=c,f*e*f^-1=b*e^-1>;
// generators/relations

Export

Character table of C2×C2.PSU3(𝔽2) in TeX

׿
×
𝔽