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C₁₃×C₂²⋊C₉ #90838

order	468
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃²×C₄×S₄ (order 864) · Inn(G) = A₄ · Out(G) = C₂×C₃²×C₄
homology	Abelianization (H₁) = C₉×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₃×C₂²⋊C₉
unambiguous semidirect product	C₂²⋊(C₉×C₁₃)

Splittings

derived, Sylow	C₂² ⋊ C₉×C₁₃
Hall	C₂²×C₁₃ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₃	A₄
commutator, hypocenter	C₂²	C₉×C₁₃
Frattini	C₃	A₄×C₁₃
Fitting, socle, prime radical	C₂²×C₃×C₁₃	C₃
cosocle	C₂²×C₃	C₃×C₁₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 234

A minimal faithful permutation representation — 2 generators on 26 points:

(2,3,4)(5,9,13,7,12,8,10,6,11)(14,15,16,17,18,19,20,21,22,23,24,25,26);(1,4)(2,3)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₁₃⋊((C₂²⋊C₉)⋊C₂), (C₂²⋊C₉)⋊Dic₁₃, (C₂²⋊C₉)⋊#261, …
maximal subgroup of	C₁₃×(C₂²⋊C₉)⋊C₂, C₁₃⋊((C₂²⋊C₉)⋊C₂), D₁₃×C₂²⋊C₉, …
maximal normal subgroup of	C₁₃×(C₂²⋊C₉)⋊C₂, C₁₃⋊((C₂²⋊C₉)⋊C₂), D₁₃×C₂²⋊C₉, …
maximal quotient of	C₁₃×Q₈⋊C₉, C₂×C₁₃×C₂²⋊C₉, C₁₃×C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 13);