C₅×D₁₁×D₅₉ #27188597

order	12980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×D₁₁×D₅₉

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₅₉ ⋊ C₂²
derived, Hall	C₁₁×C₅₉ ⋊ C₂²×C₅
Hall	C₅×C₅₉ ⋊ C₂×D₁₁
Sylow	C₅₉ ⋊ C₂×C₅×D₁₁
Hall	C₅×C₁₁ ⋊ C₂×D₅₉
Sylow	C₁₁ ⋊ C₂×C₅×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₁×D₅₉
commutator, cosocle, hypocenter	C₁₁×C₅₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₅₉	C₂²
LCS	G ▷ C₁₁×C₅₉
UCS	C₁ ◁ C₅

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

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12980, 19);

C5×D11×D59 #27188597