D₃×#491 #81736

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

D₃×#491

Splittings

Fitting	C₃×#491 ⋊ C₂
Sylow	C₃ ⋊ C₂×#491

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#491	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#491	D₃
hypocenter	C₃	C₂×#491
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1356, D₃×#1413, D₃×#1662, …
maximal normal subgroup of	D₃×#1356, D₃×#1413, D₃×#1662, …
maximal quotient of	D₃×#1836, D₃×#1935, D₃×#1960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14312);