#27190249

order	13140
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅×C₇₃ ⋊ C₄×C₉
Hall	C₅×#10620323 ⋊ C₄
Hall	#13091735 ⋊ C₉
Hall	#10620323 ⋊ Dic₅
Sylow	C₇₃ ⋊ C₉×Dic₅
Sylow	C₅ ⋊ #14581496

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇₃	C₄×C₉
prime radical, cosocle	C₅×D₇₃⋊C₃	C₂×C₃
LCS	G ▷ C₅×C₇₃
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 365

A minimal faithful permutation representation — 2 generators on 78 points:

(2,42,58,14,4,48,17,33,11,3,36,62,6,7,40,34,13,18,5,47,30,9,12,66,16,20,29,8,21,39,15,19,44,23,31,38)(10,72,53,46,25,56,59,69,51,24,35,55,27,49,64,70,37,71,26,57,41,50,32,73,52,65,60,22,63,28,43,45,67,54,61,68)(75,76)(77,78);(1,2,4,14,11,27,33,63,6,17,37,46,62,56,67,70,13,32,36,19,41,45,69,66,34,42,24,3,10,28,57,71,30,59,60,18,40,44,38,51,52,58,68,72,55,26,8,22,47,23,48,65,49,53,7,21,25,31,54,64,73,39,50,61,16,15,35,20,43,29,9,12,5)(74,75,77,78,76)

Polycyclic presentation:

2,3,2,3,5,73|a^2=c,b^3=d,e^a=e^4,f^a=f^27,f^b=f^2,f^c=f^72,f^d=f^8

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 83);