C₃×D₃×C₂₅×C₂₉ #27189381

order	13050
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₂₅×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₃×C₂₅×C₂₉
unambiguous semidirect product	C₃⋊(C₂×C₃×C₂₅×C₂₉)

Splittings

Fitting, Hall	C₃²×C₂₅×C₂₉ ⋊ C₂
derived	C₃ ⋊ C₂×C₃×C₂₅×C₂₉
Hall	C₃²×C₂₉ ⋊ C₂×C₂₅
Hall	C₃²×C₂₅ ⋊ C₂×C₂₉
Sylow	C₃² ⋊ C₂×C₂₅×C₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₂₅×C₂₉	D₃
commutator, hypocenter	C₃	C₂×C₃×C₂₅×C₂₉
Frattini	C₅	C₃×C₅×D₃×C₂₉
Fitting	C₃²×C₂₅×C₂₉	C₂
socle	C₃²×C₅×C₂₉	C₂×C₅
prime radical	C₃×C₅×D₃×C₂₉	C₅
cosocle	C₃×C₅	C₂×C₃×C₅×C₂₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₂₅×C₂₉

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 4350

A minimal faithful permutation representation — 2 generators on 60 points:

(1,2,3)(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)(36,40,48,56,60,38,44,53,59,46,41,49,57,42,51,45,54,39,47,55,50,37,43,52,58);(4,5,6)

Conjugacy classes & character table

6525 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₃×C₂₅×D₂₉, C₃×D₃×C₂₉×D₂₅, C₃×D₃×D₇₂₅, …
maximal normal subgroup of	C₃×D₃×C₂₅×D₂₉, C₃×D₃×C₂₉×D₂₅, C₃×D₃×D₇₂₅, …
maximal quotient of	C₃×Dic₃×C₂₅×C₂₉, C₂×C₃×D₃×C₂₅×C₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 13);

C3×D3×C25×C29 #27189381