#72267355

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

derived

C₂²⋊#997 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#87588
commutator, hypocenter	C₂²⋊#997	C₂×C₄
Frattini	C₂²	C₂×#87588
Fitting	C₂³×#585	D₃
socle	C₂⁴	C₂×#1020
prime radical	C₂×#11710929	C₂
cosocle	C₂×C₂²⋊#997	C₂²
LCS	G ▷ C₂²⋊#997
UCS	C₁ ◁ C₂×C₄

At least 8008 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has at least 8008 conjugacy classes of subgroups — too many to draw. Only its 94 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:

2^3,3,2^7|a^2=c,b^2=k,d^a=d^2,e^a=eik,e^d=f,f^a=efjk,f^d=ef,g^d=h,h^a=gh,h^d=gh,i^d=ij,i^f=ik,j^a=ij,j^d=i,j^e=jk,j^f=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267358
maximal normal subgroup of	#72267358

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