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C₃×#6104 #10597635

order	624
isoclinism	isoclinic class of #6104
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃×D₄×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ D₈
Sylow	C₁₃ ⋊ C₃×D₈

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂²×C₃
Frattini	C₄	C₂×C₃×D₁₃
Fitting	C₃×D₄×C₁₃	C₂
socle	C₂×C₃×C₁₃	D₄
hypercenter	C₃×D₄	D₁₃
hypocenter	C₁₃	C₃×D₈
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₄

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3)(4,5,7,9,11,13,15,16,14,12,10,8,6)(18,21)(19,24)(22,23);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6104, #11928621, #11928622, …
maximal normal subgroup of	D₃×#6104, #11928621, #11928622, …
maximal quotient of	C₃×#88047, C₃×#88049, C₃×#88066, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 72);