C₃×D₃×C₁₁×C₆₇ #27210731

order	13266
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₁×C₆₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₃×C₁₁×C₆₇
unambiguous semidirect product	C₃⋊(C₂×C₃×C₁₁×C₆₇)

Splittings

Fitting, socle, Hall	C₃²×C₁₁×C₆₇ ⋊ C₂
derived	C₃ ⋊ C₂×C₃×C₁₁×C₆₇
Hall	C₃²×C₆₇ ⋊ C₂×C₁₁
Hall	C₃²×C₁₁ ⋊ C₂×C₆₇
Sylow	C₃² ⋊ C₂×C₁₁×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₁×C₆₇	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃×C₁₁×C₆₇
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁×C₆₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₁₁×C₆₇

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 4422

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

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

Conjugacy classes & character table

6633 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₃×C₁₁×D₆₇, C₃×D₃×D₁₁×C₆₇, C₃×D₃×D₇₃₇, …
maximal normal subgroup of	C₃×D₃×C₁₁×D₆₇, C₃×D₃×D₁₁×C₆₇, C₃×D₃×D₇₃₇, …
maximal quotient of	C₃×C₁₁×Dic₃×C₆₇, C₂×C₃×D₃×C₁₁×C₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13266, 45);

C3×D3×C11×C67 #27210731