#72258996

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	H(5,𝔽₂)² ⋊ D₃
derived	#72258992 ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	#72209700
commutator, cosocle, hypocenter	#72258992	C₂
Fitting	H(5,𝔽₂)²	D₃
LCS	G ▷ #72258992
UCS	C₁ ◁ C₂²

Only the 81 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 81 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:

2,3,2^10|b^a=b^2,c^a=d,c^b=d,d^a=c,d^b=cd,e^b=f,e^c=ek,e^d=ek,f^a=ef,f^b=ef,f^d=fk,g^a=h,g^b=h,g^e=gk,h^a=g,h^b=gh,h^e=hk,h^f=hk,i^b=j,i^c=ik,i^d=ik,i^g=il,i^h=il,j^a=ij,j^b=ij,j^d=jk,j^h=jl

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259091
maximal normal subgroup of	#72259091

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