C₉×C₅⋊D₅×D₃₇ #45686015

order	33300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product

C₉×C₅⋊D₅×D₃₇

Splittings

Fitting, Hall	C₅²×C₉×C₃₇ ⋊ C₂²
derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₉
Hall	C₉×C₃₇ ⋊ C₂×C₅⋊D₅
Hall	C₅²×C₉ ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×C₉×C₅⋊D₅
Sylow	C₅² ⋊ C₂×C₉×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₉	C₅⋊D₅×D₃₇
commutator, hypocenter	C₅²×C₃₇	C₂²×C₉
Frattini	C₃	C₃×C₅⋊D₅×D₃₇
Fitting	C₅²×C₉×C₃₇	C₂²
socle, cosocle	C₃×C₅²×C₃₇	C₂²×C₃
prime radical	C₃×C₅⋊D₅×D₃₇	C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₉

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 8325

A minimal faithful permutation representation — 3 generators on 56 points:

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

Conjugacy classes & character table

2520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 288);

C9×C5⋊D5×D37 #45686015