C₁₁×#11927413 #27211889

order	13420
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₁×#11927413

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₆₁ ⋊ C₄
derived, Hall	C₅×C₆₁ ⋊ C₄×C₁₁
Hall	C₁₁×C₆₁ ⋊ AGL(1,𝔽₅)
Sylow	C₆₁ ⋊ C₁₁×AGL(1,𝔽₅)
Hall	C₅×C₁₁ ⋊ #6923
Sylow	C₅ ⋊ C₁₁×#6923

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	#11927413
commutator, hypocenter	C₅×C₆₁	C₄×C₁₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₆₁	C₄
prime radical	C₁₁×D₃₀₅	C₂
cosocle	D₃₀₅	C₂×C₁₁
LCS	G ▷ C₅×C₆₁
UCS	C₁ ◁ C₁₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3355

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 81);

C11×#11927413 #27211889