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C₃×C₄×C₁₃⋊C₃ #90847

order	468
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = C₁₃⋊C₃ · Out(G) = C₂×C₄×D₃
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₄×C₁₃⋊C₃
unambiguous semidirect product	(C₄×C₁₃)⋊C₃²
unambiguous semidirect product	(C₃×C₄×C₁₃)⋊C₃

Splittings

Fitting	C₃×C₄×C₁₃ ⋊ C₃
derived, Sylow	C₁₃ ⋊ C₃²×C₄
Hall	C₄×C₁₃ ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₃²×C₄
Frattini	C₂	C₂×C₃×C₁₃⋊C₃
Fitting	C₃×C₄×C₁₃	C₃
socle	C₂×C₃×C₁₃	C₂×C₃
prime radical	C₂×C₃×C₁₃⋊C₃	C₂
cosocle	C₂×C₁₃	C₂×C₃²
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#1055⋊C₃, C₃⋊C₈×C₁₃⋊C₃, C₃⋊(#1055⋊C₃), …
maximal normal subgroup of	C₃×#1055⋊C₃, C₃⋊C₈×C₁₃⋊C₃, C₃⋊(#1055⋊C₃), …
maximal quotient of	C₃×C₈×C₁₃⋊C₃, C₂×C₃×C₄×C₁₃⋊C₃, C₄×C₉×C₁₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 22);