C₅×C₁₁⋊#6923 #27211885

order	13420
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×C₁₁⋊#6923

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₆₁ ⋊ C₄
derived, Hall	C₁₁×C₆₁ ⋊ C₄×C₅
Hall	C₅×C₆₁ ⋊ Dic₁₁
Sylow	C₆₁ ⋊ C₅×Dic₁₁
Hall	C₅×C₁₁ ⋊ #6923
Sylow	C₁₁ ⋊ C₅×#6923

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₁₁⋊#6923
commutator, hypocenter	C₁₁×C₆₁	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₆₁	C₄
prime radical	C₅×C₁₁×D₆₁	C₂
cosocle	C₁₁×D₆₁	C₂×C₅
LCS	G ▷ C₁₁×C₆₁
UCS	C₁ ◁ C₅

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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 77);

C5×C11⋊#6923 #27211885