#27190247

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	#13091734 ⋊ C₉
Hall	#10620323 ⋊ AGL(1,𝔽₅)
Sylow	C₇₃ ⋊ C₉×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #14581484

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇₃	C₄×C₉
prime radical, cosocle	D₅×C₇₃⋊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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 81);