#72268003

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

2₋¹⁺⁸ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂²×C₂²⋊(C₂²⋊A₄)
commutator, cosocle, hypocenter	2₋¹⁺⁶	C₂²×C₃
Fitting	2₋¹⁺⁸	C₃
hypercenter	D₄	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ 2₋¹⁺⁶
UCS	C₁ ◁ C₂ ◁ D₄

140135 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 140135 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268015, #72268574, #72268576
maximal normal subgroup of	#72268015, #72268574, #72268576

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