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C₁₃×#214 #10597652

order	624
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₁₃ ⋊ SD₁₆
Sylow	C₃ ⋊ C₁₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₁₃	#67
commutator, cosocle	C₃×C₄	C₂²×C₁₃
Frattini	C₄	C₂×D₃×C₁₃
Fitting, prime radical	C₃×D₄×C₁₃	C₂
socle	C₂×C₃×C₁₃	D₄
hypercenter	D₄×C₁₃	D₃
hypocenter	C₃	C₁₃×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₄×C₁₃ ◁ D₄×C₁₃

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:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₃×#214, #11928625, #11928628, …
maximal normal subgroup of	D₁₃×#214, #11928625, #11928628, …
maximal quotient of	C₁₃×#808, C₁₃×#810, C₁₃×#832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 89);