D₁₁×C₁₃⋊AGL(1,𝔽₁₁) #44538167

order	31460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₁₁×C₁₃⋊AGL(1,𝔽₁₁)
unambiguous semidirect product	(D₁₁×C₁₁⋊C₅)⋊D₁₃

Splittings

Fitting, socle, derived, Hall	C₁₁²×C₁₃ ⋊ C₂²×C₅
Hall	C₁₁×C₁₃×C₁₁⋊C₅ ⋊ C₂²
Hall	D₁₁×D₁₄₃ ⋊ C₅
Hall	C₁₁×C₁₁⋊C₅ ⋊ C₂×D₁₃
Sylow	C₁₁² ⋊ C₂×C₅×D₁₃
Sylow	C₁₃ ⋊ D₁₁×AGL(1,𝔽₁₁)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁²×C₁₃	C₂²×C₅
LCS	G ▷ C₁₁²×C₁₃
UCS	C₁

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1430

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31460, 107);

D11×C13⋊AGL(1,𝔽11) #44538167