#72259055

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁵×D₄ ⋊ C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×A₄⋊S₄
commutator, cosocle	C₂²×A₄²	C₂³
Fitting	C₂⁵×D₄	C₃⋊D₃
socle	C₂⁶	C₂²×C₃⋊D₃
hypercenter	C₂×D₄	A₄⋊S₄
hypocenter	A₄²	#120
LCS	G ▷ C₂²×A₄² ▷ A₄²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

12862 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 12862 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

2^5,3^2,2^4|b^a=be,c^b=cd,f^c=f^2,g^c=g^2,h^c=i,h^f=i,h^g=i,i^c=h,i^f=hi,i^g=hi,j^c=hjk,j^f=k,j^g=hij,k^c=hik,k^f=jk,k^g=hk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 10, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.8*F.9*F.11);;
gap> SetConjugate(coll, 11, 6, F.10*F.11);;
gap> SetConjugate(coll, 11, 7, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);