C₂²×C₁₁⋊C₅×C₆₁ #27211882

order	13420
isoclinism	isoclinic class of C₁₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×C₁₁⋊C₅×C₆₁
unambiguous semidirect product	(C₂²×C₁₁)⋊(C₅×C₆₁)

Splittings

Fitting, socle, Hall	C₂²×C₁₁×C₆₁ ⋊ C₅
derived, Sylow	C₁₁ ⋊ C₂²×C₅×C₆₁
Hall	C₁₁×C₆₁ ⋊ C₂²×C₅
Hall	C₂²×C₁₁ ⋊ C₅×C₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₆₁	C₁₁⋊C₅
commutator, cosocle, hypocenter	C₁₁	C₂²×C₅×C₆₁
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁×C₆₁	C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂²×C₆₁

40 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:

(2,3,5,8,7)(4,6,10,11,9)(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)(75,76);(1,2,4,7,3,5,9,11,6,8,10)(73,74)

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 74);

C22×C11⋊C5×C61 #27211882