#72258287

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×H(5,𝔽₂) ⋊ D₃
derived	#72257803 ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72209700
commutator, cosocle, hypocenter	#72257803	C₂
Fitting	C₂⁴×H(5,𝔽₂)	D₃
socle	C₂⁵	#1020
LCS	G ▷ #72257803
UCS	C₁ ◁ C₂

85347 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 85347 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48

Polycyclic presentation:

2,3,2^9|b^a=b^2,c^a=d,c^b=d,d^a=c,d^b=cd,e^b=f,e^c=ek,e^d=ek,f^a=ef,f^b=ef,f^d=fk,g^a=h,g^b=h,h^a=g,h^b=gh,i^b=j,j^a=ij,j^b=ij

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258322, #72258456
maximal normal subgroup of	#72258322, #72258456
maximal quotient of	#72258529, #72258673, #72258996

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