C₅×C₃⋊D₃×C₃₇ #16809693

order	3330
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₃₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊D₃×C₃₇
unambiguous semidirect product	C₃⋊(C₅×D₃×C₃₇)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₃₇	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₃₇	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅×C₃₇

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 — 3 generators on 48 points:

(1,2,3,4,5)(7,8)(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);(9,10,11);(6,7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃⋊D₃×C₂₅×C₃₇, C₅×C₃₇×C₃⋊D₁₅, C₅²×C₃⋊D₃×C₃₇, …
maximal normal subgroup of	C₃⋊D₃×C₂₅×C₃₇, C₅²×C₃⋊D₃×C₃₇, C₅×C₇×C₃⋊D₃×C₃₇, …
maximal quotient of	C₃⋊D₃×C₂₅×C₃₇, C₅×C₃₇×C₃⋊D₁₅, C₅²×C₃⋊D₃×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3330, 33);

C5×C3⋊D3×C37 #16809693