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C₃×Dic₆₅ #11715924

order	780
isoclinism	isoclinic class of D₆₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

C₃×Dic₆₅

Splittings

derived, Hall	C₅×C₁₃ ⋊ C₃×C₄
Hall	C₃×C₅×C₁₃ ⋊ C₄
Hall	C₃×C₁₃ ⋊ Dic₅
Hall	C₃×C₅ ⋊ Dic₁₃
Sylow	C₁₃ ⋊ C₃×Dic₅
Sylow	C₅ ⋊ C₃×Dic₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₆₅
commutator, hypocenter	C₅×C₁₃	C₃×C₄
Frattini	C₂	C₃×D₆₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₁₃	C₂
cosocle	C₂×C₅×C₁₃	C₂×C₃
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₂×C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 780

A minimal faithful permutation representation — 2 generators on 25 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);(4,6,5,7)(9,12)(10,11)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10587066, C₃×#10587065, C₃⋊#10587065, …
maximal normal subgroup of	C₃×#10587066, C₃×#10587065, C₃⋊#10587065, …
maximal quotient of	C₃×#10587059, C₂×C₃×Dic₆₅, C₉×Dic₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(780, 7);