#72241425

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Splittings

Fitting

H(9,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂⁴×#1020
commutator, cosocle, hypocenter	#997	C₂⁵
Fitting	H(9,𝔽₂)	D₃
hypercenter	H(5,𝔽₂)	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Only the 710 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 710 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:

2^5,3,2^5|g^2=k,h^2=k,d^c=dk,e^b=ek,f^a=f^2,f^b=f^2,g^a=hk,g^b=hk,g^f=gh,h^a=gk,h^b=gk,h^f=g,h^g=hk,i^a=j,i^b=j,i^f=j,i^g=ik,j^a=i,j^b=i,j^f=ij,j^h=jk

Conjugacy classes & character table

163 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247427, #72252511, #72267049
maximal normal subgroup of	#72252511, #72267049

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