C₃²×C₅×D₅₃ #18012011

order	4770
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×C₅×D₅₃
unambiguous semidirect product	C₅₃⋊(C₂×C₃²×C₅)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₅	D₅₃
commutator, cosocle, hypocenter	C₅₃	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₅₃	C₂
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₃²×C₅

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)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Conjugacy classes & character table

1260 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×C₂₅×D₅₃, C₃²×C₅²×D₅₃, C₃²×C₅×D₂₆₅, …
maximal normal subgroup of	C₃²×C₂₅×D₅₃, C₃²×C₅²×D₅₃, C₃²×C₅×C₇×D₅₃
maximal quotient of	C₃²×C₂₅×D₅₃, C₃²×C₅²×D₅₃, C₃²×C₅×D₂₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 10);

C32×C5×D53 #18012011