D₃×#13357 #72255479

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

Aliases

direct product

D₃×#13357

Splittings

Fitting	C₃×#13357 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13357

Subgroups

	subgroup	G/H
center	C₂²	D₃×B(4,𝔽₂)
commutator, cosocle	C₂²×C₃×Q₈	C₂⁴
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₃×#13357	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	#13357	D₃
hypocenter	C₃	C₂×#13357
LCS	G ▷ C₂²×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #13357

4740 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)(6,11)(7,13)(8,15)(10,17)(12,20)(18,19)(21,22,23);(5,6,7,11,9,12,13,20)(8,10,16,18,15,19,14,17);(1,2,3,4)(6,10)(8,15)(11,17)(12,19)(14,16)(18,20)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#498909
maximal normal subgroup of	D₃×#498909

GAP

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

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

D3×#13357 #72255479