#72258498

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₂²×#62817 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, Fitting, cosocle, hypocenter	C₂²×#62817	C₃
socle	C₂⁴	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ C₂²×#62817
UCS	C₁ ◁ C₂²

211116 conjugacy classes of subgroups; only the 553 normal subgroups are shown.

normal subgroups only

This group has 211116 conjugacy classes of subgroups — too many to draw. Only its 553 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:

3,2^10|b^a=c,c^a=bc,d^a=e,e^a=de,f^a=g,f^c=fj,g^a=fg,g^b=gj,h^a=i,h^c=hk,i^a=hi,i^b=ik

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72258529, #72258673, #72258797, …
maximal subgroup of	#72258529, #72258614, #72258646, …
maximal normal subgroup of	#72258529, #72258614, #72258646, …

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