C₉×D₅×C₅₃ #18012004

order	4770
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₅×C₅₃
unambiguous semidirect product	C₅⋊(C₂×C₉×C₅₃)

Splittings

Fitting, Hall	C₅×C₉×C₅₃ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₉×C₅₃
Hall	C₅×C₅₃ ⋊ C₂×C₉
Hall	C₅×C₉ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉×C₅₃	D₅
commutator, hypocenter	C₅	C₂×C₉×C₅₃
Frattini	C₃	C₃×D₅×C₅₃
Fitting	C₅×C₉×C₅₃	C₂
socle	C₃×C₅×C₅₃	C₂×C₃
prime radical	C₃×D₅×C₅₃	C₃
cosocle	C₃×C₅	C₂×C₃×C₅₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₉×C₅₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 2385

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(11,14)(12,13)

Conjugacy classes & character table

1908 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×D₂₅×C₅₃, C₅×C₉×D₅×C₅₃, C₉×C₅⋊D₅×C₅₃, …
maximal normal subgroup of	C₅×C₉×D₅×C₅₃, C₇×C₉×D₅×C₅₃
maximal quotient of	C₉×D₂₅×C₅₃, C₅×C₉×D₅×C₅₃, C₉×C₅⋊D₅×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 3);

C9×D5×C53 #18012004