#72341492

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#1020
commutator, hypocenter	#997	C₂²×C₄
Frattini, socle	C₂²	C₂²×#1020
Fitting	C₂²×#585	D₃
prime radical	C₂²×#5950	C₂
cosocle	C₂×#997	C₂³
hypercenter	C₂²×C₄	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

7480 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 7480 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 64

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341494, #72341539, #72341544, …
maximal normal subgroup of	#72341494, #72341539, #72341544, …

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