A₄×C₁₉×C₅₉ #27212115

order	13452
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₉×C₅₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₉×C₅₉
unambiguous semidirect product	C₂²⋊(C₃×C₁₉×C₅₉)
unambiguous semidirect product	(C₂²×C₅₉)⋊(C₃×C₁₉)

Splittings

Fitting, socle, Hall	C₂²×C₁₉×C₅₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₉×C₅₉
Hall	C₂²×C₅₉ ⋊ C₃×C₁₉
Hall	C₂²×C₁₉ ⋊ C₃×C₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉×C₅₉	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₉×C₅₉
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉×C₅₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₉×C₅₉

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 4484

A minimal faithful permutation representation — 2 generators on 82 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(1,4)(2,3)

Conjugacy classes & character table

4484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 26);

A4×C19×C59 #27212115