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C₂×C₃×D₂₄ #64026

order	288
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₃×D₂₄

Splittings

Fitting	C₂×C₃²×C₈ ⋊ C₂
Sylow	C₃² ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	D₄
hypercenter	C₂×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589114, #10589115, #10589126, …
maximal normal subgroup of	#10589114, #10589115, #10589126, …
maximal quotient of	C₃×#4682, C₃×C₄×D₂₄, C₃×#4694, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 674);