C₉×D₅×C₃₇ #16809667

order	3330
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₅×C₃₇
unambiguous semidirect product	C₅⋊(C₂×C₉×C₃₇)

Splittings

Fitting, Hall	C₅×C₉×C₃₇ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₉×C₃₇
Hall	C₅×C₃₇ ⋊ C₂×C₉
Hall	C₅×C₉ ⋊ C₂×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₉×C₃₇	D₅
commutator, hypocenter	C₅	C₂×C₉×C₃₇
Frattini	C₃	C₃×D₅×C₃₇
Fitting	C₅×C₉×C₃₇	C₂
socle	C₃×C₅×C₃₇	C₂×C₃
prime radical	C₃×D₅×C₃₇	C₃
cosocle	C₃×C₅	C₂×C₃×C₃₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₉×C₃₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1665

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

1332 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×C₃₇×D₂₅, C₅×C₉×D₅×C₃₇, C₉×C₃₇×C₅⋊D₅, …
maximal normal subgroup of	C₅×C₉×D₅×C₃₇, C₇×C₉×D₅×C₃₇, C₉×D₅×C₁₁×C₃₇
maximal quotient of	C₉×C₃₇×D₂₅, C₅×C₉×D₅×C₃₇, C₉×C₃₇×C₅⋊D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 7);

C9×D5×C37 #16809667