C₃²×C₅×D₃₇ #16809686

order	3330
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₅ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×C₅×D₃₇

Splittings

Fitting, socle, Hall	C₃²×C₅×C₃₇ ⋊ C₂
derived, Sylow	C₃₇ ⋊ C₂×C₃²×C₅
Hall	C₃²×C₃₇ ⋊ C₂×C₅
Hall	C₅×C₃₇ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₅	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₃₇	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃²×C₅

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 1665

A minimal faithful permutation representation — 2 generators on 48 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×C₂₅×D₃₇, C₃²×C₅×D₁₈₅, C₃²×C₅²×D₃₇, …
maximal normal subgroup of	C₃²×C₂₅×D₃₇, C₃²×C₅²×D₃₇, C₃²×C₅×C₇×D₃₇, …
maximal quotient of	C₃²×C₂₅×D₃₇, C₃²×C₅×D₁₈₅, C₃²×C₅²×D₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 26);

C32×C5×D37 #16809686