C₅×D₉×C₃₇ #16809669

order	3330
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₉×C₃₇
unambiguous semidirect product	C₉⋊(C₂×C₅×C₃₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₃₇	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₅×C₃₇
Frattini	C₃	C₅×D₃×C₃₇
Fitting	C₅×C₉×C₃₇	C₂
socle	C₃×C₅×C₃₇	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₅×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);(7,14)(8,13)(9,12)(10,11)

Conjugacy classes & character table

1110 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×C₂₅×C₃₇, C₅²×D₉×C₃₇, C₅×C₃₇×D₄₅, …
maximal normal subgroup of	D₉×C₂₅×C₃₇, C₅²×D₉×C₃₇, C₅×C₇×D₉×C₃₇, …
maximal quotient of	D₉×C₂₅×C₃₇, C₅²×D₉×C₃₇, C₅×C₃₇×D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 9);

C5×D9×C37 #16809669