C₃₁⋊C₃ #789

order	93
isoclinism	isoclinic class of C₃₁⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₃₁) (order 930) · Inn(G) = G · Out(G) = C₂×C₅
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃₁⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₃₁ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₁	C₃
LCS	G ▷ C₃₁
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 31

A minimal faithful permutation representation — 2 generators on 31 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);(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:

3,31|b^a=b^25

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(93, 1);

C31⋊C3 #789