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## G = C2×CSU2(𝔽3)  order 96 = 25·3

### Direct product of C2 and CSU2(𝔽3)

Aliases: C2×CSU2(𝔽3), Q8.1D6, C22.4S4, SL2(𝔽3).1C22, C2.5(C2×S4), (C2×Q8).2S3, (C2×SL2(𝔽3)).2C2, SmallGroup(96,188)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3) — C2×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C2×CSU2(𝔽3)
 Upper central C1 — C22

Generators and relations for C2×CSU2(𝔽3)
G = < a,b,c,d,e | a2=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Character table of C2×CSU2(𝔽3)

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D size 1 1 1 1 8 6 6 12 12 8 8 8 6 6 6 6 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 -2 -2 -1 2 -2 0 0 1 -1 1 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 2 2 -1 2 2 0 0 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 -2 -2 2 -1 0 0 0 0 -1 1 1 -√2 √2 √2 -√2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ8 2 -2 -2 2 -1 0 0 0 0 -1 1 1 √2 -√2 -√2 √2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ9 2 -2 2 -2 -1 0 0 0 0 1 1 -1 √2 -√2 √2 -√2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ10 2 -2 2 -2 -1 0 0 0 0 1 1 -1 -√2 √2 -√2 √2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ11 3 3 3 3 0 -1 -1 1 1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ12 3 3 -3 -3 0 -1 1 -1 1 0 0 0 -1 -1 1 1 orthogonal lifted from C2×S4 ρ13 3 3 3 3 0 -1 -1 -1 -1 0 0 0 1 1 1 1 orthogonal lifted from S4 ρ14 3 3 -3 -3 0 -1 1 1 -1 0 0 0 1 1 -1 -1 orthogonal lifted from C2×S4 ρ15 4 -4 -4 4 1 0 0 0 0 1 -1 -1 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ16 4 -4 4 -4 1 0 0 0 0 -1 -1 1 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2

Smallest permutation representation of C2×CSU2(𝔽3)
On 32 points
Generators in S32
(1 14)(2 15)(3 16)(4 13)(5 27)(6 28)(7 25)(8 26)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 32)
(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)
(1 11 3 9)(2 10 4 12)(5 31 7 29)(6 30 8 32)(13 20 15 18)(14 19 16 17)(21 27 23 25)(22 26 24 28)
(2 11 10)(4 9 12)(5 8 30)(6 32 7)(13 17 20)(15 19 18)(22 27 26)(24 25 28)
(1 21 3 23)(2 25 4 27)(5 15 7 13)(6 20 8 18)(9 22 11 24)(10 28 12 26)(14 29 16 31)(17 30 19 32)

G:=sub<Sym(32)| (1,14)(2,15)(3,16)(4,13)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,20,15,18)(14,19,16,17)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,17,20)(15,19,18)(22,27,26)(24,25,28), (1,21,3,23)(2,25,4,27)(5,15,7,13)(6,20,8,18)(9,22,11,24)(10,28,12,26)(14,29,16,31)(17,30,19,32)>;

G:=Group( (1,14)(2,15)(3,16)(4,13)(5,27)(6,28)(7,25)(8,26)(9,17)(10,18)(11,19)(12,20)(21,29)(22,30)(23,31)(24,32), (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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,20,15,18)(14,19,16,17)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,17,20)(15,19,18)(22,27,26)(24,25,28), (1,21,3,23)(2,25,4,27)(5,15,7,13)(6,20,8,18)(9,22,11,24)(10,28,12,26)(14,29,16,31)(17,30,19,32) );

G=PermutationGroup([[(1,14),(2,15),(3,16),(4,13),(5,27),(6,28),(7,25),(8,26),(9,17),(10,18),(11,19),(12,20),(21,29),(22,30),(23,31),(24,32)], [(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)], [(1,11,3,9),(2,10,4,12),(5,31,7,29),(6,30,8,32),(13,20,15,18),(14,19,16,17),(21,27,23,25),(22,26,24,28)], [(2,11,10),(4,9,12),(5,8,30),(6,32,7),(13,17,20),(15,19,18),(22,27,26),(24,25,28)], [(1,21,3,23),(2,25,4,27),(5,15,7,13),(6,20,8,18),(9,22,11,24),(10,28,12,26),(14,29,16,31),(17,30,19,32)]])

C2×CSU2(𝔽3) is a maximal subgroup of   CSU2(𝔽3)⋊C4  Q8.D12  Q8.2D12  C23.14S4  SL2(𝔽3).D4  D4.5S4
C2×CSU2(𝔽3) is a maximal quotient of   Q8.D12  SL2(𝔽3)⋊Q8  C23.14S4

Matrix representation of C2×CSU2(𝔽3) in GL3(𝔽73) generated by

 72 0 0 0 1 0 0 0 1
,
 1 0 0 0 13 37 0 25 60
,
 1 0 0 0 24 59 0 36 49
,
 1 0 0 0 72 72 0 1 0
,
 1 0 0 0 43 13 0 43 30
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[1,0,0,0,13,25,0,37,60],[1,0,0,0,24,36,0,59,49],[1,0,0,0,72,1,0,72,0],[1,0,0,0,43,43,0,13,30] >;

C2×CSU2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C2xCSU(2,3)");
// GroupNames label

G:=SmallGroup(96,188);
// by ID

G=gap.SmallGroup(96,188);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,288,146,579,447,117,364,286,202,88]);
// Polycyclic

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

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