C₁₁×#11927414 #27211890

order	13420
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₁×#11927414

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₆₁ ⋊ C₄
derived, Hall	C₅×C₆₁ ⋊ C₄×C₁₁
Hall	C₁₁×C₆₁ ⋊ AGL(1,𝔽₅)
Sylow	C₆₁ ⋊ C₁₁×AGL(1,𝔽₅)
Hall	C₅×C₁₁ ⋊ Dic₆₁
Sylow	C₅ ⋊ C₁₁×Dic₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	#11927414
commutator, hypocenter	C₅×C₆₁	C₄×C₁₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₆₁	C₄
prime radical	D₅×C₁₁×C₆₁	C₂
cosocle	D₅×C₆₁	C₂×C₁₁
LCS	G ▷ C₅×C₆₁
UCS	C₁ ◁ C₁₁

24 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)(63,66,65,64)(67,68,69,70,71,72,73,74,75,76,77);(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)(62,63,64,66,65)

Conjugacy classes & character table

1375 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 82);

C11×#11927414 #27211890