C₅×D₉×C₅₃ #18012006

order	4770
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₉×C₅₃
unambiguous semidirect product	C₉⋊(C₂×C₅×C₅₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₅₃	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₅×C₅₃
Frattini	C₃	C₅×D₃×C₅₃
Fitting	C₅×C₉×C₅₃	C₂
socle	C₃×C₅×C₅₃	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₅×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);(7,14)(8,13)(9,12)(10,11)

Conjugacy classes & character table

1590 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×C₂₅×C₅₃, C₅²×D₉×C₅₃, C₅×C₅₃×D₄₅, …
maximal normal subgroup of	D₉×C₂₅×C₅₃, C₅²×D₉×C₅₃, C₅×C₇×D₉×C₅₃
maximal quotient of	D₉×C₂₅×C₅₃, C₅²×D₉×C₅₃, C₅×C₅₃×D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 5);

C5×D9×C53 #18012006