#45686151

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₃₇ ⋊ C₄×C₉
Hall	C₅²⋊#66526 ⋊ C₄
Hall	#16814877 ⋊ C₉
Sylow	C₃₇ ⋊ #1043⋊C₉
Sylow	C₅² ⋊ AGL(1,𝔽₃₇)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₃₇	C₄×C₉
prime radical, cosocle	C₅⋊(C₅⋊(D₃₇⋊C₃))	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 555

A minimal faithful permutation representation — 2 generators on 52 points:

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

Polycyclic presentation:

2,3,2,3,5^2,37|a^2=c,b^3=d,e^a=e^2,f^a=f^2,g^a=g^31,e^b=ef,f^b=e^2f^3,g^b=g^16,e^c=e^4,f^c=f^4,g^c=g^36,g^d=g^26

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 424);