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C₉×D₆₅ #11924049

order	1170
isoclinism	isoclinic class of D₆₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₆₅
unambiguous semidirect product	C₅⋊(C₉×D₁₃)

Splittings

Fitting, Hall	C₅×C₉×C₁₃ ⋊ C₂
derived, Hall	C₅×C₁₃ ⋊ C₂×C₉
Hall	C₉×C₁₃ ⋊ D₅
Hall	C₅×C₉ ⋊ D₁₃
Sylow	C₁₃ ⋊ C₉×D₅
Sylow	C₅ ⋊ C₉×D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₆₅
commutator, hypocenter	C₅×C₁₃	C₂×C₉
Frattini	C₃	C₃×D₆₅
Fitting	C₅×C₉×C₁₃	C₂
socle, cosocle	C₃×C₅×C₁₃	C₂×C₃
prime radical	C₃×D₆₅	C₃
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₉

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)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×#63122, C₉×#63121, C₉⋊#63121, …
maximal normal subgroup of	C₉×#63122, C₉×#63121, C₉⋊#63121, …
maximal quotient of	C₉×Dic₆₅, C₂×C₉×D₆₅, C₉×C₁₁×D₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1170, 7);