C₆₁×C₁₁⋊AGL(1,𝔽₅) #27211891

order	13420
isoclinism	isoclinic class of C₁₁⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₁×C₁₁⋊AGL(1,𝔽₅)
unambiguous semidirect product	C₁₁⋊(AGL(1,𝔽₅)×C₆₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₆₁ ⋊ C₄
derived, Hall	C₅×C₁₁ ⋊ C₄×C₆₁
Hall	C₁₁×C₆₁ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₆₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ AGL(1,𝔽₅)×C₆₁
Sylow	C₅ ⋊ Dic₁₁×C₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₆₁	C₁₁⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₁₁	C₄×C₆₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₆₁	C₄
prime radical	D₅×C₁₁×C₆₁	C₂
cosocle	D₅×C₁₁	C₂×C₆₁
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₆₁

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:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 83);

C61×C11⋊AGL(1,𝔽5) #27211891