C₅×C₉×D₂₃ #14523387

order	2070
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₉×D₂₃
unambiguous semidirect product	C₂₃⋊(C₂×C₅×C₉)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₉	D₂₃
commutator, hypocenter	C₂₃	C₂×C₅×C₉
Frattini	C₃	C₃×C₅×D₂₃
Fitting	C₅×C₉×C₂₃	C₂
socle	C₃×C₅×C₂₃	C₂×C₃
prime radical	C₃×C₅×D₂₃	C₃
cosocle	C₃×C₂₃	C₂×C₃×C₅
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₅×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);(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

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

Relationships

maximal subgroup of	C₉×D₅×D₂₃, C₅×D₉×D₂₃, D₂₃×D₄₅, …
maximal normal subgroup of	C₉×D₅×D₂₃, C₅×D₉×D₂₃, D₂₃×D₄₅, …
maximal quotient of	C₅×C₉×Dic₂₃, C₂×C₅×C₉×D₂₃, C₅×C₇×C₉×D₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2070, 2);

C5×C9×D23 #14523387