Finite Group Databasebeta

← Prev (#85865) Next (#85867) →

D₃×#518 #85866

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₃×#518 ⋊ C₂
Sylow	C₃ ⋊ C₂×#518

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator	C₂×C₃	C₂⁴×C₄
Fitting	C₃×#518	C₂
socle, cosocle	C₂²×C₃	C₂⁵
prime radical	C₂²×D₃×D₄	C₂
hypercenter	#518	D₃
hypocenter	C₃	C₂×#518
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #518

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1828, D₃×#1829, D₃×#1856, …
maximal normal subgroup of	D₃×#1828, D₃×#1829, D₃×#1856, …
maximal quotient of	D₃×#2273, D₃×#2274, D₃×#2275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18442);