C₃²×D₅×C₅₃ #18012012

order	4770
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₅×C₅₃
unambiguous semidirect product	C₅⋊(C₂×C₃²×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₅₃ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃²×C₅₃
Hall	C₅×C₅₃ ⋊ C₂×C₃²
Hall	C₃²×C₅ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₅₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₅₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₅₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 2385

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3)(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);(4,5,6)(8,11)(9,10)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×D₂₅×C₅₃, C₃²×C₅×D₅×C₅₃, C₃²×C₅⋊D₅×C₅₃, …
maximal normal subgroup of	C₃²×C₅×D₅×C₅₃, C₃²×C₇×D₅×C₅₃
maximal quotient of	C₃²×D₂₅×C₅₃, C₃²×C₅×D₅×C₅₃, C₃²×C₅⋊D₅×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 11);

C32×D5×C53 #18012012