C₁₁×C₃⋊D₆₉ #17986926

order	4554
isoclinism	isoclinic class of C₃⋊D₆₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₁₁×C₃⋊D₆₉
unambiguous semidirect product	C₃⋊(C₁₁×D₆₉)

Splittings

Fitting, socle, Hall	C₃²×C₁₁×C₂₃ ⋊ C₂
derived, Hall	C₃²×C₂₃ ⋊ C₂×C₁₁
Hall	C₁₁×C₂₃ ⋊ C₃⋊D₃
Hall	C₃²×C₁₁ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₁₁×C₃⋊D₃
Sylow	C₃² ⋊ C₁₁×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	C₃⋊D₆₉
commutator, cosocle, hypocenter	C₃²×C₂₃	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁×C₂₃	C₂
LCS	G ▷ C₃²×C₂₃
UCS	C₁ ◁ C₁₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 2277

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5,6,7,8,9,10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(36,37)(39,40);(12,13,15,17,19,21,23,25,27,29,31,33,34,32,30,28,26,24,22,20,18,16,14)(38,39,40);(35,36,37)

Conjugacy classes & character table

1155 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₁⋊C₅×C₃⋊D₆₉, C₅×C₁₁×C₃⋊D₆₉, C₁₁×C₃⋊D₃₄₅, …
maximal normal subgroup of	C₁₁⋊C₅×C₃⋊D₆₉, C₅×C₁₁×C₃⋊D₆₉, C₇×C₁₁×C₃⋊D₆₉
maximal quotient of	C₅×C₁₁×C₃⋊D₆₉, C₁₁×C₃⋊D₃₄₅, C₇×C₁₁×C₃⋊D₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4554, 28);

C11×C3⋊D69 #17986926