C₃²×D₂₆₅ #18012013

order	4770
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₂₆₅
unambiguous semidirect product	C₅⋊(C₃²×D₅₃)
unambiguous semidirect product	C₅₃⋊(C₃²×D₅)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₅₃ ⋊ C₂
derived, Hall	C₅×C₅₃ ⋊ C₂×C₃²
Hall	C₃²×C₅₃ ⋊ D₅
Sylow	C₅₃ ⋊ C₃²×D₅
Hall	C₃²×C₅ ⋊ D₅₃
Sylow	C₅ ⋊ C₃²×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₂₆₅
commutator, cosocle, hypocenter	C₅×C₅₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₅₃	C₂
LCS	G ▷ C₅×C₅₃
UCS	C₁ ◁ C₃²

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)(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

1206 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×D₁₃₂₅, C₃²×C₅×D₂₆₅, C₃²×C₅⋊D₂₆₅, …
maximal normal subgroup of	C₃²×C₅×D₂₆₅, C₃²×C₇×D₂₆₅
maximal quotient of	C₃²×D₁₃₂₅, C₃²×C₅×D₂₆₅, C₃²×C₅⋊D₂₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 12);

C32×D265 #18012013