#27211855

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	#16809823 ⋊ 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	#20269516	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:

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