C₅×C₉×D₃₇ #16809666

order	3330
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×C₉×D₃₇

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₉	D₃₇
commutator, hypocenter	C₃₇	C₂×C₅×C₉
Frattini	C₃	C₃×C₅×D₃₇
Fitting	C₅×C₉×C₃₇	C₂
socle	C₃×C₅×C₃₇	C₂×C₃
prime radical	C₃×C₅×D₃₇	C₃
cosocle	C₃×C₃₇	C₂×C₃×C₅
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₅×C₉

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);(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×C₂₅×D₃₇, C₅²×C₉×D₃₇, C₅×C₉×D₁₈₅, …
maximal normal subgroup of	C₉×C₂₅×D₃₇, C₅²×C₉×D₃₇, C₅×C₇×C₉×D₃₇, …
maximal quotient of	C₉×C₂₅×D₃₇, C₅²×C₉×D₃₇, C₅×C₉×D₁₈₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 6);

C5×C9×D37 #16809666