#72233786

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#7630442 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#85645
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₂²×C₃
Frattini, socle	C₂³	C₂²×C₂²⋊A₄
Fitting	#7630442	C₃
hypercenter	C₂²	#85645
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂ ◁ C₂²

8431 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 8431 conjugacy classes of subgroups — too many to draw. Only its 52 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|f^2=j,g^2=j,b^a=bj,d^c=de,e^c=d,f^b=fh,f^c=fg,f^e=fj,g^b=ghi,g^c=f,g^d=gj,g^f=gj,h^c=i,i^c=hi

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241278, #72241321, #72258646, …
maximal normal subgroup of	#72241278, #72241321, C₂×#72233786
maximal quotient of	#72233869, #72267096, C₂×#72233786

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