#72257803

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₂⁴×H(5,𝔽₂) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, Fitting, cosocle, hypocenter	C₂⁴×H(5,𝔽₂)	C₃
socle	C₂⁵	C₂²⋊A₄
LCS	G ▷ C₂⁴×H(5,𝔽₂)
UCS	C₁ ◁ C₂

153287 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

normal subgroups only

This group has 153287 conjugacy classes of subgroups — too many to draw. Only its 537 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) = 48

Polycyclic presentation:

3,2^9|b^a=bc,c^a=b,d^a=de,d^c=dj,e^a=d,e^b=ej,f^a=g,g^a=fg,h^a=ghi,h^e=hj,i^a=fgh,i^d=ij

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72258287, #72258322, #72258456, …
maximal subgroup of	#72258206, #72258287, #72258302, …
maximal normal subgroup of	#72258206, #72258287, #72258302, …
maximal quotient of	#72258498, #72258992

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