D₃×#13261 #72258658

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

D₃×#13261

Splittings

Fitting	C₃×#13261 ⋊ C₂
Sylow	C₃ ⋊ C₂×#13261

Subgroups

	subgroup	G/H
center	C₂²	D₃×#459
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13261	C₂
socle	C₂²×C₃	C₂×#459
prime radical	C₂×D₃×#493	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13261	D₃
hypocenter	C₃	C₂×#13261
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #13261

3220 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:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495850
maximal normal subgroup of	D₃×#495850

GAP

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

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

D3×#13261 #72258658