#72258302

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72250267 ⋊ C₃
derived	C₂⁴×H(5,𝔽₂) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72215614
commutator, cosocle, hypocenter	C₂⁴×H(5,𝔽₂)	C₂×C₃
Frattini, socle	C₂⁵	C₂×C₂²⋊A₄
Fitting	#72250267	C₃
LCS	G ▷ C₂⁴×H(5,𝔽₂)
UCS	C₁ ◁ C₂

81643 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 81643 conjugacy classes of subgroups — too many to draw. Only its 51 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|c^a=cg,c^b=d,d^a=dh,d^b=cd,e^a=ei,e^b=f,e^c=ek,e^d=ek,f^a=fj,f^b=ef,f^d=fk,g^b=h,h^b=gh,i^b=j,j^b=ij

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258322
maximal normal subgroup of	#72258322
maximal quotient of	#72258646

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