C₃×C₅⋊#11925890 #31827646

order	18150
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₃×C₅⋊#11925890
unambiguous semidirect product	C₅⋊(C₃×#11925890)

Splittings

Fitting, socle	C₃×C₅×C₁₁² ⋊ D₅
derived, Hall	C₅×#10597245 ⋊ C₂×C₃
Hall	C₃×C₅×#10597245 ⋊ C₂
Hall	C₃×C₁₁² ⋊ C₅⋊D₅
Sylow	C₁₁² ⋊ C₃×C₅⋊D₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₅⋊#11925890
commutator, cosocle, hypocenter	C₅×#10597245	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₁²	D₅
LCS	G ▷ C₅×#10597245
UCS	C₁ ◁ C₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 330

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,7,9,22,17,5,6,14,16,23,18,8,10,19,21,12,24,15,11,13,25,20)(27,28)(29,30);(26,27,29,30,28);(6,9,16,18,11)(7,12,19,14,8)(10,17,24,25,21)(13,22,20,23,15)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×D₅×#11925890, C₃×#11925890⋊D₅, C₃×#27130929, …
maximal normal subgroup of	C₃×D₅×#11925890, C₃×#11925890⋊D₅, C₃×#27130929, …
maximal quotient of	C₃×#27130863, C₂×C₃×C₅⋊#11925890

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18150, 130);

C3×C5⋊#11925890 #31827646