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C₂×#4680 #78439

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

Splittings

C₃ ⋊ C₂×#498

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×#498
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690449, #10690451, #10694402, …
maximal normal subgroup of	#10690449, #10690451, #10694402, …
maximal quotient of	C₂×#69764, C₂×#69769, C₂×#69777, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11015);