#72267656

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

solvable

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#577210 ⋊ C₃
derived	#2840 ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#3549⋊C₃
commutator, hypocenter	#2840	C₃×C₄
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	#577210	C₃
prime radical	C₂×#73292	C₂
cosocle	C₂×#2840	C₂×C₃
LCS	G ▷ #2840
UCS	C₁ ◁ C₂²

1053 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1053 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) = 96

Polycyclic presentation:

2,3,2^8|a^2=c,d^2=h,e^2=i,f^2=j,g^2=j,b^a=bhj,d^a=dhj,d^b=de,e^a=eij,e^b=dij,e^d=ej,f^a=fij,f^b=g,f^d=fhj,f^e=fhij,g^a=ghj,g^b=fgj,g^d=gi,g^e=gh,g^f=gj,h^b=hij,i^b=h

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267658
maximal quotient of	#72267657

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