#72268583

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

solvable

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	#87524
commutator, cosocle	#5948	C₂³
Frattini	C₄	C₂²×#1020
Fitting	C₂²×#585	D₃
socle	C₂³	C₂×GL(2,ℤ/4)
hypercenter	#41	#1020
hypocenter	C₂²⋊SL(2,𝔽₃)	C₂×D₄
LCS	G ▷ #5948 ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

7223 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 7223 conjugacy classes of subgroups — too many to draw. Only its 61 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:

2^4,3,2^5|b^2=j,c^2=dj,d^2=j,h^2=j,i^2=j,b^a=bj,c^b=cd,d^b=dj,e^c=e^2,f^c=fg,f^e=g,g^e=fg,h^c=hj,h^e=i,i^c=hi,i^e=hij,i^h=ij

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72268584

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