C₅×C₁₉×D₆₅ #27134704

order	12350
isoclinism	isoclinic class of D₆₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₉×D₆₅
unambiguous semidirect product	C₅⋊(C₅×C₁₉×D₁₃)
unambiguous semidirect product	C₁₃⋊(C₅×D₅×C₁₉)

Splittings

Fitting, socle, Hall	C₅²×C₁₃×C₁₉ ⋊ C₂
derived	C₅×C₁₃ ⋊ C₂×C₅×C₁₉
Hall	C₅²×C₁₉ ⋊ D₁₃
Hall	C₅²×C₁₃ ⋊ C₂×C₁₉
Hall	C₁₃×C₁₉ ⋊ C₅×D₅
Sylow	C₅² ⋊ C₁₉×D₁₃
Sylow	C₁₃ ⋊ C₅×D₅×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₁₉	D₆₅
commutator, cosocle, hypocenter	C₅×C₁₃	C₂×C₅×C₁₉
Frattini	C₁	G
Fitting, socle	C₅²×C₁₃×C₁₉	C₂
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₅×C₁₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 2470

A minimal faithful permutation representation — 2 generators on 42 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,16,17,18)(20,21)(22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(1,2,4,6,8,10,12,13,11,9,7,5,3)(19,20,22,23,21)

Conjugacy classes & character table

3230 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₁₉×D₆₅, C₅×D₅×C₁₉×D₁₃, C₅×D₆₅⋊D₁₉, …
maximal normal subgroup of	C₅×D₁₉×D₆₅, C₅×D₅×C₁₉×D₁₃, C₅×D₆₅⋊D₁₉, …
maximal quotient of	C₅×C₁₉×Dic₆₅, C₂×C₅×C₁₉×D₆₅, C₃×C₅×C₁₉×D₆₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12350, 15);

C5×C19×D65 #27134704