C₃×C₃⋊#13091735 #27190296

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

Aliases

direct product	C₃×C₃⋊#13091735
unambiguous semidirect product	C₃⋊(C₃×#13091735)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₇₃ ⋊ C₄
derived	C₃×C₅×C₇₃ ⋊ C₃×C₄
Hall	C₃²×C₇₃ ⋊ Dic₅
Hall	C₅×C₇₃ ⋊ C₃×Dic₃
Sylow	C₇₃ ⋊ C₃×Dic₁₅
Hall	C₃²×C₅ ⋊ #64408
Sylow	C₃² ⋊ #13091735
Sylow	C₅ ⋊ C₃×C₃⋊#64408

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊#13091735
commutator, hypocenter	C₃×C₅×C₇₃	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇₃	C₄
prime radical	C₃²×C₅×D₇₃	C₂
cosocle	C₃×C₅×D₇₃	C₂×C₃
LCS	G ▷ C₃×C₅×C₇₃
UCS	C₁ ◁ C₃

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 130);

C3×C3⋊#13091735 #27190296