C₁₁×C₆₇⋊C₃ #14555642

order	2211
isoclinism	isoclinic class of C₆₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 737

A minimal faithful permutation representation — 2 generators on 78 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₁×C₆₇⋊C₃, C₁₁×D₆₇⋊C₃, D₁₁×C₆₇⋊C₃, …
maximal normal subgroup of	C₂×C₁₁×C₆₇⋊C₃, C₁₁×D₆₇⋊C₃, D₁₁×C₆₇⋊C₃, …
maximal quotient of	C₂×C₁₁×C₆₇⋊C₃, C₅×C₁₁×C₆₇⋊C₃, C₇×C₁₁×C₆₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2211, 3);

C11×C67⋊C3 #14555642