C₆₁×Dic₁₁⋊C₅ #27211830

order	13420
isoclinism	isoclinic class of AGL(1,𝔽₁₁)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₁×Dic₁₁⋊C₅
unambiguous semidirect product	Dic₁₁⋊(C₅×C₆₁)
unambiguous semidirect product	(C₁₁⋊C₅)⋊(C₄×C₆₁)

Splittings

derived, Sylow	C₁₁ ⋊ C₄×C₅×C₆₁
Hall	C₁₁⋊C₅×C₆₁ ⋊ C₄
Hall	Dic₁₁×C₆₁ ⋊ C₅
Hall	C₁₁×C₆₁ ⋊ C₄×C₅
Hall	C₁₁⋊C₅ ⋊ C₄×C₆₁
Hall	Dic₁₁ ⋊ C₅×C₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₁	AGL(1,𝔽₁₁)
commutator, hypocenter	C₁₁	C₄×C₅×C₆₁
Frattini	C₂	C₆₁×AGL(1,𝔽₁₁)
Fitting, socle	C₂×C₁₁×C₆₁	C₂×C₅
prime radical	C₂×C₁₁⋊C₅×C₆₁	C₂
cosocle	C₂×C₁₁	C₂×C₅×C₆₁
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂×C₆₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 2684

A minimal faithful permutation representation — 2 generators on 76 points:

(2,5,6,10,9,3,4,7,8,11)(12,13,14,15)(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);(1,2,5,8,6,9,11,7,10,4,3)

Conjugacy classes & character table

1342 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 22);

C61×Dic11⋊C5 #27211830