C₄₉⋊C₉ #89199

order	441
isoclinism	isoclinic class of C₄₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₄₉⋊C₉

Splittings

derived, Sylow

C₄₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₄₉⋊C₃
commutator, hypocenter	C₄₉	C₉
Frattini, socle, prime radical	C₃×C₇	ΓL(1,𝔽₈)
Fitting, cosocle	C₃×C₄₉	C₃
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₃

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 441

A minimal faithful permutation representation — 2 generators on 58 points:

(2,4,8)(3,7,15)(5,11,23)(6,13,20)(9,19,17)(10,22,31)(12,26,44)(14,30,18)(16,33,32)(21,38,37)(24,40,34)(25,43,29)(27,45,47)(28,41,35)(36,39,46)(42,49,48)(50,51,53,52,54,56,55,57,58);(1,45,30,40,25,24,48,7,6,5,49,18,17,26,15,23,10,9,27,39,19,33,43,29,28,11,37,38,3,2,47,46,8,42,41,16,14,4,12,22,35,34,32,31,13,21,20,44,36)

Polycyclic presentation:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₉⋊C₉, C₂×C₄₉⋊C₉, C₉×C₄₉⋊C₃, …
maximal normal subgroup of	D₄₉⋊C₉, C₂×C₄₉⋊C₉, C₉×C₄₉⋊C₃, …
maximal quotient of	C₂×C₄₉⋊C₉, C₄₉⋊C₂₇, C₃×C₄₉⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(441, 1);

C49⋊C9 #89199