C₃₇×D₄₅ #16809671

order	3330
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×D₄₅
unambiguous semidirect product	C₅⋊(D₉×C₃₇)
unambiguous semidirect product	C₉⋊(D₅×C₃₇)

Splittings

Fitting, Hall	C₅×C₉×C₃₇ ⋊ C₂
derived, Hall	C₅×C₉ ⋊ C₂×C₃₇
Hall	C₉×C₃₇ ⋊ D₅
Hall	C₅×C₃₇ ⋊ D₉
Sylow	C₉ ⋊ D₅×C₃₇
Sylow	C₅ ⋊ D₉×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	D₄₅
commutator, cosocle, hypocenter	C₅×C₉	C₂×C₃₇
Frattini	C₃	D₁₅×C₃₇
Fitting	C₅×C₉×C₃₇	C₂
socle	C₃×C₅×C₃₇	D₃
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ 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);(39,46)(40,45)(41,44)(42,43)(48,51)(49,50)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×D₂₂₅, C₅×C₃₇×D₄₅, C₃₇×C₅⋊D₄₅, …
maximal normal subgroup of	C₅×C₃₇×D₄₅, C₇×C₃₇×D₄₅, C₁₁×C₃₇×D₄₅
maximal quotient of	C₃₇×D₂₂₅, C₅×C₃₇×D₄₅, C₃₇×C₅⋊D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 11);

C37×D45 #16809671