SL(2,𝔽₃)×GU(2,𝔽₃) #72276722

order	2304
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

solvable

Aliases

direct product

SL(2,𝔽₃)×GU(2,𝔽₃)

Splittings

derived

Q₈×SL(2,𝔽₃) ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	A₄×S₄
commutator, hypocenter	Q₈×SL(2,𝔽₃)	C₃×C₄
Fitting	Q₈×#41	C₃×D₃
socle	C₂²	A₄×SL(2,ℤ/4)
prime radical	SL(2,𝔽₃)×#41⋊C₃	C₂
cosocle	Q₈×#41⋊C₃	C₂×C₃
LCS	G ▷ Q₈×SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄

356 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

SL(2,𝔽₃)×GL(2,𝔽₅)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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