#72252743

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂⁴×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	C₂²×C₃×C₂²⋊A₄
commutator, cosocle	C₃×H(5,𝔽₂)	C₂²×C₃²
Fitting	C₃²×H(7,𝔽₂)	C₃
hypercenter	C₃²×D₄	C₂²⋊A₄
hypocenter	H(5,𝔽₂)	C₂²×H(3,𝔽₃)
LCS	G ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×D₄

6415 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 6415 conjugacy classes of subgroups — too many to draw. Only its 128 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:

2^2,3^3,2^5|a^2=j,f^2=j,g^2=j,b^a=bj,d^c=de,f^c=g,g^c=fgj,g^f=gj,h^c=fhi,h^g=hj,i^c=ghj,i^f=ij

Conjugacy classes & character table

235 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, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);