C₃×C₅×C₃⋊#64408 #27190290

order	13140
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×C₃⋊#64408
unambiguous semidirect product	C₃⋊(C₃×C₅×#64408)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₇₃ ⋊ C₄
derived	C₃×C₇₃ ⋊ C₃×C₄×C₅
Hall	C₃²×C₇₃ ⋊ C₄×C₅
Hall	C₅×C₇₃ ⋊ C₃×Dic₃
Sylow	C₇₃ ⋊ C₃×C₅×Dic₃
Hall	C₃²×C₅ ⋊ #64408
Sylow	C₃² ⋊ C₅×#64408

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	C₃⋊#64408
commutator, hypocenter	C₃×C₇₃	C₃×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇₃	C₄
prime radical	C₃²×C₅×D₇₃	C₂
cosocle	C₃×D₇₃	C₂×C₃×C₅
LCS	G ▷ C₃×C₇₃
UCS	C₁ ◁ C₃×C₅

56 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 124);

C3×C5×C3⋊#64408 #27190290