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C₃×#6122 #10597721

order	624
isoclinism	isoclinic class of #6118
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₁₃ ⋊ M₁₆
Sylow	C₁₃ ⋊ C₃×M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×#261
commutator	C₂×C₁₃	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×#261
Fitting, prime radical	C₂²×C₃×C₁₃	C₄
socle	C₂×C₃×C₁₃	C₂×C₄
cosocle	Dic₁₃	C₂²×C₃
hypercenter	C₂²×C₃	#261
hypocenter	C₁₃	C₃×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

40 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)(4,5,8,11,15,9,13,14,6,10,16,12,7)(18,21)(22,24);(5,6,7,9)(8,11,12,16)(10,14,15,13)(17,18,19,22,20,21,23,24)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11928221, #11928222, C₃×#88113, …
maximal normal subgroup of	#11928221, #11928222, C₃×#88113, …
maximal quotient of	C₃×#88110, C₃×#88112, C₃×C₂²⋊#1057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 158);