#72228560

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10586097 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂²	C₂³×C₂²⋊A₄
commutator, cosocle	C₂×H(5,𝔽₂)	C₂³×C₃
Fitting	#10586097	C₃
hypercenter	#127	C₂²⋊A₄
hypocenter	H(5,𝔽₂)	C₂×C₃×D₄
LCS	G ▷ C₂×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ #127

19987 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 19987 conjugacy classes of subgroups — too many to draw. Only its 159 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:

2^3,3,2^6|c^2=j,f^2=j,g^2=j,b^a=be,c^a=cj,c^b=cj,f^d=fg,g^d=f,g^f=gj,h^d=i,h^f=hj,h^g=hj,i^d=hi,i^f=ij

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228568, #72246076, #72246181, …
maximal normal subgroup of	#72228568, #72246076, #72246181, …
maximal quotient of	C₂×#72228560

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