D₃×#499 #81915

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₃×#499

Splittings

Fitting	C₃×#499 ⋊ C₂
Sylow	C₃ ⋊ C₂×#499

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#499	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#499	D₃
hypocenter	C₃	C₂×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

360 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1363, #10706455, D₃×#1366, …
maximal normal subgroup of	D₃×#1363, #10706455, D₃×#1366, …
maximal quotient of	D₃×#1846, D₃×#1941, D₃×#1942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14491);