D₃×#13344 #72248148

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#13344

Splittings

Fitting	C₃×#13344 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13344

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#13344	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×D₃×B(4,𝔽₂)	C₂
hypercenter	#13344	D₃
hypocenter	C₃	C₂×#13344
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #13344

5612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(22,23);(2,4)(5,7)(6,11,12,18)(8,13)(9,14)(10,16,17,19)(15,20)(21,22,23);(5,6,8,12)(7,11)(9,10)(13,18)(14,19,20,16)(15,17);(1,2,3,4)(6,10,12,17)(7,13)(11,16,18,19)(14,20)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#498885, D₃×#498888
maximal normal subgroup of	D₃×#498885, D₃×#498888

GAP

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

gap> G := Group((22,23), (2,4)(5,7)(6,11,12,18)(8,13)(9,14)(10,16,17,19)(15,20)(21,22,23), (5,6,8,12)(7,11)(9,10)(13,18)(14,19,20,16)(15,17), (1,2,3,4)(6,10,12,17)(7,13)(11,16,18,19)(14,20));

D3×#13344 #72248148