#72252244

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#516323 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#87588⋊C₂
commutator, cosocle	C₂×C₂²⋊#997	C₂²
Frattini, socle	C₂⁴	C₂×#1020
Fitting	#516323	D₃
hypocenter	C₂²⋊#997	D₄
LCS	G ▷ C₂×C₂²⋊#997 ▷ C₂²⋊#997
UCS	C₁ ◁ C₂²

8850 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 8850 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^3,3,2^7|b^2=c,g^2=k,h^2=k,b^a=bc,d^a=d^2,e^a=f,e^d=f,f^a=e,f^d=ef,g^a=fh,g^b=gj,g^d=h,g^f=gk,h^a=eg,h^b=hi,h^d=ghk,h^e=hk,h^g=hk,i^a=j,i^d=ij,j^a=i,j^d=i

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252245, #72267097, #72267153, …
maximal normal subgroup of	#72252245, #72267097, #72267153, …
maximal quotient of	#72258556, C₂×#72252244

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