#72267943

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#1020
commutator, hypocenter	C₂²⋊SL(2,𝔽₃)	C₂²×C₄
Frattini	C₂²	C₂²×#1020
Fitting	C₂⁴×#41	D₃
socle	C₂⁴	C₂²×S₄
prime radical	C₂²×#5948	C₂
cosocle	C₂×C₂²⋊SL(2,𝔽₃)	C₂³
hypercenter	C₂²×C₄	#1020
LCS	G ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

7844 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 7844 conjugacy classes of subgroups — too many to draw. Only its 113 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=d,c^2=j,h^2=j,i^2=j,b^a=bj,c^a=cj,e^a=e^2,f^a=fg,f^e=g,g^e=fg,h^a=fghi,h^e=i,i^a=gij,i^e=hij,i^h=ij

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267944
maximal normal subgroup of	#72267944
maximal quotient of	#72267948

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