#72268648

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#87524
commutator	#5948	C₂×C₄
Frattini	C₂×C₄	C₂×#1020
Fitting	C₂²×#585	D₃
socle	C₂³	#5432
prime radical	#11710926	C₂
cosocle	C₂×#5948	C₂²
hypercenter	#41	#1020
hypocenter	C₂²⋊SL(2,𝔽₃)	C₂²⋊C₄
LCS	G ▷ #5948 ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

6058 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 6058 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 96

Polycyclic presentation:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268761
maximal quotient of	#72268761

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