Finite Group Databasebeta

← Prev (#81008) Next (#81010) →

D₃×#475 #81009

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

Aliases

Splittings

Fitting	C₃×#475 ⋊ C₂
Sylow	C₃ ⋊ C₂×#475

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#475	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂²×D₃	C₂×D₄
hypercenter	#475	D₃
hypocenter	C₃	C₂×#475
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

416 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:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1643, D₃×#1640, D₃×#1651, …
maximal normal subgroup of	D₃×#1643, D₃×#1640, D₃×#1651, …
maximal quotient of	D₃×#1803, D₃×#1865, D₃×#1919, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 13585);