#72246775

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#87588
commutator, cosocle, hypocenter	C₂²⋊#997	C₂²
Fitting	C₂³×H(5,𝔽₂)	D₃
socle	C₂³	C₂×#1020
hypercenter	C₂²	#87588
LCS	G ▷ C₂²⋊#997
UCS	C₁ ◁ C₂ ◁ C₂²

7658 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 7658 conjugacy classes of subgroups — too many to draw. Only its 39 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:

2^2,3,2^7|b^2=j,b^a=bj,c^b=c^2,d^b=defj,d^c=e,e^b=efgj,e^c=de,f^b=g,f^c=g,f^d=fj,f^e=fj,g^b=f,g^c=fg,g^e=gj,h^b=i,h^c=i,h^d=hj,h^e=hj,i^b=h,i^c=hi,i^e=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258000, #72258360, #72258673, …
maximal normal subgroup of	#72258000, #72258360, #72267352, …
maximal quotient of	#72247274, #72267356, #72340867

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