#72252733

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁶×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×H(7,𝔽₂) ⋊ C₃
Sylow	H(7,𝔽₂) ⋊ H(3,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂⁴×C₃×A₄
commutator, cosocle	C₃×Q₈	C₂⁴×C₃²
Fitting	C₃²×H(7,𝔽₂)	C₃
hypercenter	C₃²×2₋¹⁺⁴	A₄
hypocenter	Q₈	C₂⁴×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×2₋¹⁺⁴

7252 conjugacy classes of subgroups; only the 673 normal subgroups are shown.

normal subgroups only

This group has 7252 conjugacy classes of subgroups — too many to draw. Only its 673 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) = 144

Polycyclic presentation:

2^4,3^3,2^3|c^2=j,d^2=j,h^2=j,i^2=j,b^a=bj,d^c=dj,f^e=fg,h^f=hi,i^f=h,i^h=ij

Conjugacy classes & character table

331 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);