C₃×D₃×D₅×C₇₃ #27190298

order	13140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×D₃×D₅×C₇₃
unambiguous semidirect product	(C₅×D₃)⋊(C₂×C₃×C₇₃)
unambiguous semidirect product	(C₃×D₅)⋊(C₂×C₃×C₇₃)
unambiguous semidirect product	D₁₅⋊(C₂×C₃×C₇₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₇₃ ⋊ C₂²
derived	C₃×C₅ ⋊ C₂²×C₃×C₇₃
Hall	C₃²×C₇₃ ⋊ C₂×D₅
Hall	C₅×C₇₃ ⋊ C₂×C₃×D₃
Hall	C₃²×C₅ ⋊ C₂²×C₇₃
Sylow	C₃² ⋊ C₂×D₅×C₇₃
Sylow	C₅ ⋊ C₂×C₃×D₃×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇₃	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₃×C₇₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇₃	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₇₃

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 2190

A minimal faithful permutation representation — 2 generators on 84 points:

(2,3)(4,5)(6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(1,2,4,5,3)(10,11)

Conjugacy classes & character table

2628 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 132);

C3×D3×D5×C73 #27190298