C₉×D₂₆₅ #18012005

order	4770
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₂₆₅
unambiguous semidirect product	C₅⋊(C₉×D₅₃)
unambiguous semidirect product	C₅₃⋊(C₉×D₅)

Splittings

Fitting, Hall	C₅×C₉×C₅₃ ⋊ C₂
derived, Hall	C₅×C₅₃ ⋊ C₂×C₉
Hall	C₉×C₅₃ ⋊ D₅
Sylow	C₅₃ ⋊ C₉×D₅
Hall	C₅×C₉ ⋊ D₅₃
Sylow	C₅ ⋊ C₉×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₂₆₅
commutator, hypocenter	C₅×C₅₃	C₂×C₉
Frattini	C₃	C₃×D₂₆₅
Fitting	C₅×C₉×C₅₃	C₂
socle, cosocle	C₃×C₅×C₅₃	C₂×C₃
prime radical	C₃×D₂₆₅	C₃
LCS	G ▷ C₅×C₅₃
UCS	C₁ ◁ C₉

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);(11,14)(12,13)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×D₁₃₂₅, C₅×C₉×D₂₆₅, C₉×C₅⋊D₂₆₅, …
maximal normal subgroup of	C₅×C₉×D₂₆₅, C₇×C₉×D₂₆₅
maximal quotient of	C₉×D₁₃₂₅, C₅×C₉×D₂₆₅, C₉×C₅⋊D₂₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 4);

C9×D265 #18012005