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C₉×D₅×C₁₃ #11924048

order	1170
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₅×C₁₃
unambiguous semidirect product	C₅⋊(C₂×C₉×C₁₃)

Splittings

Fitting, Hall	C₅×C₉×C₁₃ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₉×C₁₃
Hall	C₅×C₁₃ ⋊ C₂×C₉
Hall	C₅×C₉ ⋊ C₂×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉×C₁₃	D₅
commutator, hypocenter	C₅	C₂×C₉×C₁₃
Frattini	C₃	C₃×D₅×C₁₃
Fitting	C₅×C₉×C₁₃	C₂
socle	C₃×C₅×C₁₃	C₂×C₃
prime radical	C₃×D₅×C₁₃	C₃
cosocle	C₃×C₅	C₂×C₃×C₁₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₉×C₁₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 585

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27);(11,14)(12,13)

Conjugacy classes & character table

468 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×C₁₃×AGL(1,𝔽₅), C₉×#63120, C₁₃×C₉⋊AGL(1,𝔽₅), …
maximal normal subgroup of	C₉×C₁₃×AGL(1,𝔽₅), C₉×#63120, C₁₃×C₉⋊AGL(1,𝔽₅), …
maximal quotient of	C₉×C₁₃×Dic₅, C₂×C₉×D₅×C₁₃, C₉×D₅×C₁₁×C₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1170, 6);