D₃×C₇₃ #89143

order	438
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = D₃×C₈×C₉ (order 432) · Inn(G) = D₃ · Out(G) = C₈×C₉
homology	Abelianization (H₁) = C₂×C₇₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₇₃
unambiguous semidirect product	C₃⋊(C₂×C₇₃)

Splittings

Fitting, socle, Hall	C₃×C₇₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₇₃
Frattini	C₁	G
Fitting, socle	C₃×C₇₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₇₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 219

A minimal faithful permutation representation — 2 generators on 76 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76);(1,2,3)

Conjugacy classes & character table

219 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇₃, C₂×D₃×C₇₃, D₉×C₇₃, …
maximal normal subgroup of	D₃×D₇₃, C₂×D₃×C₇₃, D₃×C₇₃⋊C₃, …
maximal quotient of	Dic₃×C₇₃, C₂×D₃×C₇₃, D₉×C₇₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(438, 3);

D3×C73 #89143

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

D₃×C₇₃ #89143