#72252878

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₂×M₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#13049789
commutator, socle	C₂×C₃⁴	C₂²×C₈
Fitting	C₂²×C₃⁴	C₂×C₈
prime radical	C₂²×C₃⋊(C₃⋊(C₃⋊D₃))	C₈
cosocle	C₃⋊(C₃⋊#622)	C₂³
hypercenter	C₂²	#13049789
hypocenter	C₃⁴	C₂×M₃₂
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

846 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 59 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:

2^6,3^4|a^2=d,c^2=f,d^2=e,e^2=f,b^a=bf,g^a=gh,g^d=g^2h,g^e=g^2,h^a=g,h^d=gh,h^e=h^2,i^a=gh^2ij,i^c=gh^2i^2,i^d=ghi^2,i^e=g^2hi,j^a=hij^2,j^c=hj^2,j^d=g^2hj^2,j^e=h^2j

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252886
maximal normal subgroup of	#72252886

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