#72227403

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#1421181 ⋊ C₃
derived	C₂²×Q₈² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#85645
commutator, cosocle, hypocenter	C₂²×Q₈²	C₂×C₃
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	#1421181	C₃
LCS	G ▷ C₂²×Q₈²
UCS	C₁ ◁ C₂²

1635 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1635 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 192

Polycyclic presentation:

2,3,2^8|c^2=j,d^2=j,e^2=j,f^2=j,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=dj,e^b=ef,e^c=ei,e^d=ej,f^b=e,f^c=fij,f^d=fi,f^e=fj,g^b=h,h^b=gh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227546
maximal normal subgroup of	#72227546
maximal quotient of	#72227406, #72267885, #72268092

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