C₃₁⋊C₉ #63292

order	279
isoclinism	isoclinic class of C₃₁⋊C₃
automorphism	Aut(G) = C₃×AGL(1,𝔽₃₁) (order 2790) · Inn(G) = C₃₁⋊C₃ · Out(G) = C₂×C₃×C₅
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃₁⋊C₉

Splittings

derived, Sylow

C₃₁ ⋊ C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃₁⋊C₃
commutator, hypocenter	C₃₁	C₉
Fitting, socle, prime radical, cosocle	C₃×C₃₁	C₃
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₃

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 279

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

3^2,31|a^3=b,c^a=c^25

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₁⋊C₉, C₂×C₃₁⋊C₉, C₉×C₃₁⋊C₃, …
maximal normal subgroup of	D₃₁⋊C₉, C₂×C₃₁⋊C₉, C₉×C₃₁⋊C₃, …
maximal quotient of	C₂×C₃₁⋊C₉, C₃₁⋊C₂₇, C₃×C₃₁⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(279, 1);

C31⋊C9 #63292