#72209641

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	#62817 ⋊ D₃
derived	#11710730 ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	#87588
commutator, cosocle, hypocenter	#11710730	C₂
Fitting	#62817	D₃
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂²

2531 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 2531 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^8|g^2=ij,h^2=ij,b^a=b^2,c^a=d,c^b=cd,d^a=c,d^b=c,e^a=f,e^b=f,f^a=e,f^b=ef,g^a=cfh,g^b=h,g^c=gi,g^f=gj,h^a=deg,h^b=ghij,h^d=hi,h^e=hj,h^g=hij

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209643, #72209832, #72209837, …
maximal normal subgroup of	#72209832, #72209837, #72209887, …
maximal quotient of	#72209763, #72209773, C₂×#72209641, …
outer automorphism group of	C₂²×C₄×D₃

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