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D₃×#398 #77945

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

Aliases

Splittings

Fitting	C₃×#398 ⋊ C₂
Sylow	C₃ ⋊ C₂×#398

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#398	C₂
prime radical	C₂³×D₃	C₂³
hypercenter	#398	D₃
hypocenter	C₃	C₂×#398
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #398

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1346, #10705871, D₃×#1609, …
maximal normal subgroup of	D₃×#1346, #10705871, D₃×#1609, …
maximal quotient of	D₃×#1441, D₃×#1442, D₃×#1444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 10521);