#27211846

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₁₁×C₆₁ ⋊ C₄×C₅
Hall	#16809822 ⋊ C₄
Hall	C₁₁⋊#6923 ⋊ C₅
Sylow	C₆₁ ⋊ Dic₁₁⋊C₅
Sylow	C₁₁ ⋊ #6923⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₁×C₆₁	C₄×C₅
prime radical	#20269515	C₂
cosocle	C₁₁×D₆₁	C₂×C₅
LCS	G ▷ C₁₁×C₆₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 671

A minimal faithful permutation representation — 2 generators on 72 points:

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

Polycyclic presentation:

2,5,2,11,61|a^2=c,d^a=d^10,e^a=e^50,d^b=d^4,e^b=e^20,e^c=e^60

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 38);