Copied to
clipboard

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.A45C2, C8○D43S3, C2.8(C4×S4), C4.28(C2×S4), C4○D4.9D6, Q8.4(C4×S3), C4.6S4.2C2, C4.A4.11C22, SL2(𝔽3).3(C2×C4), 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 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6, C8, C8 [×4], C2×C4 [×3], D4 [×3], Q8, Q8, Dic3, C12, D6, C42, C2×C8 [×3], M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊C8, C24, SL2(𝔽3), C4×S3, C4×C8, C4≀C2 [×2], C8.C4, C8○D4, C8○D4, C4○D8, S3×C8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C8○D8, U2(𝔽3), C8.A4, C4.6S4, CU2(𝔽3)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, 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++++++++
imageC1C2C2C2C4C4S3D6C4×S3CU2(𝔽3)S4C2×S4C4×S4CU2(𝔽3)
kernelCU2(𝔽3)U2(𝔽3)C8.A4C4.6S4CSU2(𝔽3)GL2(𝔽3)C8○D4C4○D4Q8C1C8C4C2C1
# 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

׿
×
𝔽