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D₃×#421 #78991

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Splittings

Fitting	C₃×#421 ⋊ C₂
Sylow	C₃ ⋊ C₂×#421

Subgroups

	subgroup	G/H
center	C₄	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#421	C₂
socle	C₂×C₃	C₂²×C₂²⋊C₄
prime radical	C₂×D₃×#41	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#421	D₃
hypocenter	C₃	C₂×#421
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1794, #10794512, D₃×#1795, …
maximal normal subgroup of	D₃×#1794, #10794512, D₃×#1795, …
maximal quotient of	D₃×#1478, D₃×#1479, D₃×#1477, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11567);