#27211853

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	#16809824 ⋊ C₄
Hall	C₁₁×#6923 ⋊ C₅
Hall	#6923 ⋊ C₁₁⋊C₅
Sylow	C₆₁ ⋊ C₄×C₁₁⋊C₅
Sylow	C₁₁ ⋊ #6923⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₁×C₆₁	C₄×C₅
prime radical	#20269514	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:

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

Polycyclic presentation:

2,5,2,11,61|a^2=c,e^a=e^50,d^b=d^4,e^b=e^9,e^c=e^60

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 45);