#72241491

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₃² ⋊ D₃²
derived	C₃²×A₄² ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×A₄²	C₂²
Fitting, socle	C₂⁴×C₃²	D₃²
LCS	G ▷ C₃²×A₄²
UCS	C₁

1888 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1888 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 14 points:

(1,3,2)(5,8,6,7)(10,11)(12,13);(3,4)(7,8)(9,10,11)(13,14);(6,7,8)

Polycyclic presentation:

2^2,3^4,2^4|c^a=c^2,c^b=c^2,d^a=d^2,d^b=c^2d,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^c=h,g^d=gh,h^a=gh,h^b=gh,h^c=gh,h^d=g,i^a=gi,i^b=gi,i^c=gi,i^d=j,j^a=hij,j^c=ghj,j^d=ij

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×S₄×C₃⋊S₄, D₃×#13181790, S₄×#89101
maximal normal subgroup of	D₃×S₄×C₃⋊S₄, D₃×#13181790, S₄×#89101

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", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.3^2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.7*F.9);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);