C₉×D₁₁₅ #14523389

order	2070
isoclinism	isoclinic class of D₁₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₁₁₅
unambiguous semidirect product	C₅⋊(C₉×D₂₃)
unambiguous semidirect product	C₂₃⋊(C₉×D₅)

Splittings

Fitting, Hall	C₅×C₉×C₂₃ ⋊ C₂
derived, Hall	C₅×C₂₃ ⋊ C₂×C₉
Hall	C₉×C₂₃ ⋊ D₅
Hall	C₅×C₉ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₉×D₅
Sylow	C₅ ⋊ C₉×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₁₁₅
commutator, hypocenter	C₅×C₂₃	C₂×C₉
Frattini	C₃	C₃×D₁₁₅
Fitting	C₅×C₉×C₂₃	C₂
socle, cosocle	C₃×C₅×C₂₃	C₂×C₃
prime radical	C₃×D₁₁₅	C₃
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 1035

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(11,14)(12,13)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×D₅×D₂₃, D₉×D₁₁₅, D₄₅⋊D₂₃, …
maximal normal subgroup of	C₉×D₅×D₂₃, D₉×D₁₁₅, D₄₅⋊D₂₃, …
maximal quotient of	C₉×Dic₁₁₅, C₂×C₉×D₁₁₅, C₇×C₉×D₁₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2070, 4);

C9×D115 #14523389