C₂×C₃×#10593663 #72269031

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³×C₃

solvable

Aliases

direct product

C₂×C₃×#10593663

Splittings

Fitting

C₂×C₃²×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²×C₃	#64269
commutator, cosocle	C₃×#41⋊C₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×C₃⋊S₄
Fitting	C₂×C₃²×H(5,𝔽₂)	D₃
socle	C₂²×C₃²	GL(2,ℤ/4)
hypercenter	C₂×C₃×Q₈	C₃⋊S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂×C₃×D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₈

2042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72252958

GAP

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

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

C2×C3×#10593663 #72269031