#72344078

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10574873 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×#120⋊C₃
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂⁴×C₃
Frattini	C₂³	C₂⁴×A₄
Fitting	#10574873	C₃
hypercenter	C₂×D₄	#120⋊C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂ ◁ C₂×D₄

7923 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 7923 conjugacy classes of subgroups — too many to draw. Only its 218 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:

2^4,3,2^5|d^2=j,f^2=j,g^2=j,c^b=cj,d^a=dj,f^a=fhi,f^b=fhi,f^c=fhi,f^e=g,g^a=gh,g^b=gh,g^c=gh,g^e=fgj,g^f=gj,h^e=ij,h^f=hj,i^e=hij,i^f=ij,i^g=ij

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72344080
maximal normal subgroup of	#72344080
maximal quotient of	#72344128

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