D₃×#401 #78429

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₃×#401

Splittings

Fitting	C₃×#401 ⋊ C₂
Sylow	C₃ ⋊ C₂×#401

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#401	C₂
prime radical	C₂³×D₃	C₂³
hypercenter	#401	D₃
hypocenter	C₃	C₂×#401
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #401

516 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×#401, D₃×#2401, D₃×#2643, …
maximal normal subgroup of	C₂×D₃×#401, D₃×#2401, D₃×#2643, …
maximal quotient of	D₃×#1443, #10717049, Dic₃×#401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11005);

D3×#401 #78429

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

D₃×#401 #78429