C₂×C₃²×C₅×C₅₃ #18012010

order	4770
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 9984
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

abelian

C₂×C₃²×C₅×C₅₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 4770

A minimal faithful permutation representation — 2 generators on 66 points:

(1,2)(3,4,5)(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);(6,7,8)

Conjugacy classes & character table

4770 conjugacy classes. The character table is too large to display.

Relationships

socle of	C₂×C₃²×C₂₅×C₅₃
maximal subgroup of	C₂×C₃²×C₂₅×C₅₃, C₂×C₃²×C₅²×C₅₃, C₃²×C₅×D₅×C₅₃, …
maximal normal subgroup of	C₂×C₃²×C₂₅×C₅₃, C₂×C₃²×C₅²×C₅₃, C₂×C₃²×C₅×C₇×C₅₃, …
maximal quotient of	C₂×C₃²×C₂₅×C₅₃, C₂×C₃²×C₅²×C₅₃, C₃²×C₅×D₅×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 9);

C2×C32×C5×C53 #18012010