#72260190

order	2232
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃₁ ⋊ C₃×D₃
derived	A₄×C₃₁ ⋊ C₂×C₃
Sylow	C₃₁ ⋊ C₃×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	A₄×C₃₁	C₂×C₃
Fitting, socle	C₂²×C₃₁	C₃×D₃
LCS	G ▷ A₄×C₃₁
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 124

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2,3^2,31,2^2|c^a=c^2,d^a=d^30,d^b=d^25,e^a=ef,e^c=f,f^c=ef

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260216, S₄×D₃₁⋊C₃
maximal normal subgroup of	#72260216, S₄×D₃₁⋊C₃

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 3, 31, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 4, 2, F.4^25);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);