#72213894

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#1842 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₄	#72994
commutator	C₄×C₄²⋊C₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×S₄
Fitting	C₂²×#1842	D₃
socle	C₂⁴	C₂×C₄⋊S₄
prime radical	C₂×#11712792	C₂
cosocle	C₂×C₄×C₄²⋊C₃	C₂³
hypercenter	C₂²×C₄	#857⋊C₂
hypocenter	C₄²⋊C₃	#416
LCS	G ▷ C₄×C₄²⋊C₃ ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

3456 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 3456 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^4|a^2=d,b^2=e,e^2=f,h^2=j,i^2=k,b^a=bdef,c^a=cfk,e^a=ef,g^a=g^2j,g^b=gjk,g^c=gk,g^d=gj,h^a=hik,h^b=hj,h^g=ij,i^a=ik,i^b=ik,i^g=hi,j^a=jk,j^g=k,k^g=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214062, #72218219
maximal normal subgroup of	#72218219

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