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C₂×#4543 #71734

order	384
isoclinism	isoclinic class of #4537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₃ ⋊ C₂×#330

Subgroups

	subgroup	G/H
center	C₂³	#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₂×C₃×#125	C₂
socle, prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×#125	D₃
hypocenter	C₃	C₂×#330
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#4543, Dic₃×#330, #10686113, …
maximal normal subgroup of	C₄×#4543, Dic₃×#330, #10686113, …
maximal quotient of	C₂×#67638, #10683159, #10683413, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4310);