#72245233

order	3072
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

derived

#11710730 ⋊ C₄

Subgroups

	subgroup	G/H
center, socle	C₂²	#11715850
commutator, hypocenter	#11710730	C₄
Frattini, hypercenter	C₂³	#87588
Fitting	C₂×#62817	D₃
prime radical, cosocle	C₂×#11710730	C₂
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂² ◁ C₂³

9794 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 9794 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 32

Polycyclic presentation:

2^2,3,2^8|a^2=b,c^a=c^2,d^a=e,d^c=e,e^a=d,e^c=de,f^a=g,f^c=g,f^d=fk,f^e=fj,g^a=f,g^c=fg,g^d=gjk,g^e=gk,h^a=hi,h^c=i,h^f=hk,i^c=hi,i^f=ik,i^g=ik,j^a=jk

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246350, #72246591, C₄×#72209772, …
maximal normal subgroup of	#72246350, #72246591, C₄×#72209772
maximal quotient of	#72258530

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