#72258952

order	12288
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#6499874 ⋊ D₃
derived	#72258498 ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₂	#72258456
commutator, cosocle, hypocenter	#72258498	C₂²
Frattini	C₂⁴	C₂×#87588
Fitting	C₂²×#6499874	D₃
socle	C₂³	#72209653
hypercenter	C₂²	#72258354
LCS	G ▷ #72258498
UCS	C₁ ◁ C₂ ◁ C₂²

Only the 59 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 59 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^10|b^2=m,c^a=c^2,d^a=e,d^c=e,e^a=d,e^c=de,f^a=g,f^c=g,g^a=f,g^c=fg,h^a=i,h^b=hj,h^c=i,h^e=hl,i^a=h,i^b=ik,i^c=hi,i^d=il,j^a=k,j^c=k,j^e=jm,k^a=j,k^c=jk,k^d=km,l^b=lm

Conjugacy classes & character table

106 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", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.13);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);