D₅×D₁₁×C₅₉ #27188604

order	12980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅×D₁₁×C₅₉
unambiguous semidirect product	(D₅×C₁₁)⋊(C₂×C₅₉)
unambiguous semidirect product	(C₅×D₁₁)⋊(C₂×C₅₉)
unambiguous semidirect product	D₅₅⋊(C₂×C₅₉)

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₅₉ ⋊ C₂²
derived, Hall	C₅×C₁₁ ⋊ C₂²×C₅₉
Hall	C₁₁×C₅₉ ⋊ C₂×D₅
Hall	C₅×C₅₉ ⋊ C₂×D₁₁
Sylow	C₁₁ ⋊ C₂×D₅×C₅₉
Sylow	C₅ ⋊ C₂×D₁₁×C₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅₉	D₅×D₁₁
commutator, cosocle, hypocenter	C₅×C₁₁	C₂²×C₅₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₅₉	C₂²
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₅₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 3245

A minimal faithful permutation representation — 2 generators on 75 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,16,14)(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);(1,2,4,6,8,10,11,9,7,5,3)(13,14)(15,16)

Conjugacy classes & character table

1652 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12980, 26);

D5×D11×C59 #27188604