C₃²×C₅⋊D₅×D₃₇ #45686398

order	33300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₅

Aliases

direct product

C₃²×C₅⋊D₅×D₃₇

Splittings

Fitting, socle, Hall	C₃²×C₅²×C₃₇ ⋊ C₂²
derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₃²
Hall	C₃²×C₃₇ ⋊ C₂×C₅⋊D₅
Hall	C₃²×C₅² ⋊ C₂×D₃₇
Sylow	C₃₇ ⋊ C₂×C₃²×C₅⋊D₅
Sylow	C₅² ⋊ C₂×C₃²×D₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	C₅⋊D₅×D₃₇
commutator, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₂²
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃²

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 8325

A minimal faithful permutation representation — 3 generators on 53 points:

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

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, 671);

C32×C5⋊D5×D37 #45686398