#45686179

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	#16814878 ⋊ C₉
Hall	#66526 ⋊ #1043
Sylow	C₃₇ ⋊ C₉×#1043
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

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 925

A minimal faithful permutation representation — 3 generators on 47 points:

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

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^6,g^b=g^16,e^c=e^4,f^c=f^4,g^c=g^36,g^d=g^26

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 452);