C₃×D₆₉ #88029

order	414
isoclinism	isoclinic class of D₆₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₆₉
unambiguous semidirect product	C₃⋊(C₃×D₂₃)
unambiguous semidirect product	C₂₃⋊(C₃×D₃)

Splittings

Fitting, socle, Hall	C₃²×C₂₃ ⋊ C₂
derived	C₃×C₂₃ ⋊ C₂×C₃
Sylow	C₂₃ ⋊ C₃×D₃
Sylow	C₃² ⋊ D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₆₉
commutator, cosocle, hypocenter	C₃×C₂₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₂₃	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₃

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 138

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29);(4,5,6)(7,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×D₂₃, D₃×D₆₉, (C₃⋊D₃)⋊D₂₃, …
maximal normal subgroup of	C₃×D₃×D₂₃, D₃×D₆₉, (C₃⋊D₃)⋊D₂₃, …
maximal quotient of	C₃×Dic₆₉, C₂×C₃×D₆₉, C₃×D₂₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(414, 7);

C3×D69 #88029

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×D₆₉ #88029