#45685967

order	33300
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Hall	C₅²×C₃₇ ⋊ C₄×C₉
Hall	C₅²⋊#66524 ⋊ C₄
Hall	#16814877 ⋊ C₉
Sylow	C₃₇ ⋊ #1043⋊C₉
Sylow	C₅² ⋊ #13050613

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	#26012838
commutator, hypocenter	C₅²×C₃₇	C₄×C₉
Fitting, socle	C₃×C₅²×C₃₇	C₃×C₄
prime radical, cosocle	C₃×C₅⋊D₁₈₅	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1665

A minimal faithful permutation representation — 2 generators on 61 points:

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

Polycyclic presentation:

2,3,2,5^2,37,3|a^2=c,b^3=g,d^a=d^3,e^a=e^3,f^a=f^6,d^b=de,e^b=d^2e^3,f^b=f^26,d^c=d^4,e^c=e^4,f^c=f^36

Conjugacy classes & character table

267 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 240);