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G = CU2(F3)  order 192 = 26·3

Conformal unitary group on F32

non-abelian, soluble

Aliases: CU2(F3), C8.8S4, U2(F3):7C2, GL2(F3):3C4, CSU2(F3):3C4, C8.A4:5C2, C8oD4:3S3, C2.8(C4xS4), C4.28(C2xS4), C4oD4.9D6, Q8.4(C4xS3), C4.6S4.2C2, C4.A4.11C22, SL2(F3).3(C2xC4), SmallGroup(192,963)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — CU2(F3)
C1C2Q8SL2(F3)C4.A4C4.6S4 — CU2(F3)
SL2(F3) — CU2(F3)
C1C8

Generators and relations for CU2(F3)
 G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >

Subgroups: 215 in 64 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2xC4, D4, Q8, Q8, Dic3, C12, D6, C42, C2xC8, M4(2), D8, SD16, Q16, C4oD4, C4oD4, C3:C8, C24, SL2(F3), C4xS3, C4xC8, C4wrC2, C8.C4, C8oD4, C8oD4, C4oD8, S3xC8, CSU2(F3), GL2(F3), C4.A4, C8oD8, U2(F3), C8.A4, C4.6S4, CU2(F3)
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C4xS3, S4, C2xS4, C4xS4, CU2(F3)

Smallest permutation representation of CU2(F3)
On 32 points
Generators in S32
(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 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)
(9 27 23)(10 28 24)(11 29 17)(12 30 18)(13 31 19)(14 32 20)(15 25 21)(16 26 22)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)

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

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

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

32 conjugacy classes

class 1 2A2B2C 3 4A4B4C···4G4H 6 8A8B8C8D8E···8J8K8L12A12B24A24B24C24D
order12223444···44688888···888121224242424
size116128116···612811116···61212888888

32 irreducible representations

dim11111122223334
type++++++++
imageC1C2C2C2C4C4S3D6C4xS3CU2(F3)S4C2xS4C4xS4CU2(F3)
kernelCU2(F3)U2(F3)C8.A4C4.6S4CSU2(F3)GL2(F3)C8oD4C4oD4Q8C1C8C4C2C1
# reps11112211282244

Matrix representation of CU2(F3) in GL2(F17) generated by

20
02
,
130
104
,
137
04
,
42
1512
,
24
1215
G:=sub<GL(2,GF(17))| [2,0,0,2],[13,10,0,4],[13,0,7,4],[4,15,2,12],[2,12,4,15] >;

CU2(F3) in GAP, Magma, Sage, TeX

{\rm CU}_2({\mathbb F}_3)
% in TeX

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

G:=SmallGroup(192,963);
// by ID

G=gap.SmallGroup(192,963);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,36,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=d^3=e^2=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^4*c,e*d*e=d^-1>;
// generators/relations

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