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## G = C2×C2.PSU3(𝔽2)  order 288 = 25·32

### Direct product of C2 and C2.PSU3(𝔽2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×C2.PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C2.PSU3(𝔽2) — C2×C2.PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — C2×C2.PSU3(𝔽2)
 Upper central C1 — C22

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 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C size 1 1 1 1 9 9 9 9 8 18 18 18 18 18 18 18 18 18 18 18 18 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 linear of order 2 ρ8 1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ9 1 -1 1 -1 -1 -1 1 1 1 i -i i -i -1 1 -1 1 -i i -i i -1 1 -1 linear of order 4 ρ10 1 -1 1 -1 -1 -1 1 1 1 -i i -i i -1 1 -1 1 i -i i -i -1 1 -1 linear of order 4 ρ11 1 -1 -1 1 1 -1 -1 1 1 i -i -i i 1 -1 -1 1 i -i -i i 1 -1 -1 linear of order 4 ρ12 1 -1 -1 1 1 -1 -1 1 1 -i i i -i 1 -1 -1 1 -i i i -i 1 -1 -1 linear of order 4 ρ13 1 -1 1 -1 -1 -1 1 1 1 i i -i i 1 -1 1 -1 -i i -i -i -1 1 -1 linear of order 4 ρ14 1 -1 1 -1 -1 -1 1 1 1 -i -i i -i 1 -1 1 -1 i -i i i -1 1 -1 linear of order 4 ρ15 1 -1 -1 1 1 -1 -1 1 1 i i i -i -1 1 1 -1 i -i -i -i 1 -1 -1 linear of order 4 ρ16 1 -1 -1 1 1 -1 -1 1 1 -i -i -i i -1 1 1 -1 -i i i i 1 -1 -1 linear of order 4 ρ17 2 -2 2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 orthogonal lifted from D4 ρ18 2 -2 -2 2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ19 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 symplectic lifted from Q8, Schur index 2 ρ20 2 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 symplectic lifted from Q8, Schur index 2 ρ21 8 -8 -8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2.PSU3(𝔽2) ρ22 8 8 -8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 orthogonal lifted from C2×PSU3(𝔽2) ρ23 8 8 8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from PSU3(𝔽2) ρ24 8 -8 8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 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)

 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 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 12 1 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 1 12
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 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 12 0 0 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 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 0 0 0 12 0 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 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 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 0 0

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

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