C₄₉⋊C₃ #3858

order	147
isoclinism	isoclinic class of C₄₉⋊C₃
automorphism	Aut(G) = #72215772 (order 2058) · Inn(G) = G · Out(G) = C₂×C₇
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₉⋊C₃

Splittings

Fitting, derived, Sylow

C₄₉ ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉	C₃
Frattini, socle	C₇	ΓL(1,𝔽₈)
LCS	G ▷ C₄₉
UCS	C₁

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 49

A minimal faithful permutation representation — 2 generators on 49 points:

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

Polycyclic presentation:

3,7^2|b^7=c,b^a=b^2c^4,c^a=c^2

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₉⋊C₃, C₂×C₄₉⋊C₃, C₃×C₄₉⋊C₃, …
maximal normal subgroup of	D₄₉⋊C₃, C₂×C₄₉⋊C₃, C₃×C₄₉⋊C₃, …
maximal quotient of	C₂×C₄₉⋊C₃, C₄₉⋊C₉, C₃×C₄₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(147, 1);

C49⋊C3 #3858