D₅₅×D₆₁ #27211898

order	13420
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅₅×D₆₁
unambiguous semidirect product	(C₅×D₆₁)⋊D₁₁
unambiguous semidirect product	(C₁₁×D₆₁)⋊D₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₁×C₆₁ ⋊ C₂²
Hall	C₁₁×C₆₁ ⋊ C₂×D₅
Hall	C₅×C₆₁ ⋊ C₂×D₁₁
Sylow	C₆₁ ⋊ C₂×D₅₅
Hall	C₅×C₁₁ ⋊ C₂×D₆₁
Sylow	C₁₁ ⋊ D₅×D₆₁
Sylow	C₅ ⋊ D₁₁×D₆₁

Subgroups

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

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3355

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 90);

D55×D61 #27211898