D₃×#13055 #72254229

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₃×#13055

Splittings

Fitting	C₃×#13055 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13055

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator	C₂×C₃×Q₈	C₂³×C₄
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₃×#13055	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×D₃×#454	C₂
cosocle	C₂²×C₃×Q₈	C₂⁴
hypercenter	#13055	D₃
hypocenter	C₃	C₂×#13055
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #13055

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#494622
maximal normal subgroup of	D₃×#494622

GAP

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

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

D3×#13055 #72254229