#27211844

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	#16809821 ⋊ 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	#20269517	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:

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