D₃×C₄₉ #64414

order	294
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₃×D₃×C₇ (order 252) · Inn(G) = D₃ · Out(G) = C₂×C₃×C₇
homology	Abelianization (H₁) = C₂×C₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₄₉
unambiguous semidirect product	C₃⋊(C₂×C₄₉)

Splittings

Fitting, Hall	C₃×C₄₉ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₄₉	D₃
commutator, hypocenter	C₃	C₂×C₄₉
Frattini	C₇	D₃×C₇
Fitting	C₃×C₄₉	C₂
socle, cosocle	C₃×C₇	C₂×C₇
prime radical	D₃×C₇	C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₄₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 147

A minimal faithful permutation representation — 2 generators on 52 points:

(2,3)(4,5,7,10,14,19,25,6,8,11,15,20,26,32,9,12,16,21,27,33,38,13,17,22,28,34,39,43,18,23,29,35,40,44,47,24,30,36,41,45,48,50,31,37,42,46,49,51,52);(1,2,3)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₄₉, C₂×D₃×C₄₉, D₉×C₄₉, …
maximal normal subgroup of	D₃×D₄₉, C₂×D₃×C₄₉, D₃×C₄₉⋊C₃, …
maximal quotient of	Dic₃×C₄₉, C₂×D₃×C₄₉, D₉×C₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 3);

D3×C49 #64414

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

D₃×C₄₉ #64414