#72209832

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#6499874 ⋊ D₃
derived	#11710730 ⋊ C₂²

Subgroups

	subgroup	G/H
center, socle	C₂	#72209653
commutator, cosocle, hypocenter	#11710730	C₂²
Frattini	C₂⁴	C₂×#1020
Fitting	#6499874	D₃
hypercenter	C₂²	#87588⋊C₂
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂ ◁ C₂²

3539 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 3539 conjugacy classes of subgroups — too many to draw. Only its 23 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:

2^2,3,2^8|f^2=j,g^2=j,c^b=c^2,d^a=dhi,d^b=e,d^c=e,e^a=eh,e^b=d,e^c=de,f^a=fhi,f^b=dfgj,f^c=g,f^d=fj,g^a=gh,g^b=deg,g^c=fgj,g^d=gj,g^e=gj,g^f=gj,h^b=hi,h^c=i,h^g=hk,i^c=hi,i^f=ik,j^a=jk

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209833, #72215896, C₃×#72209832, …
maximal normal subgroup of	#72217104, C₂×#72209832, C₃×#72209832, …
maximal quotient of	C₂×#72209832, #72227696, #72227778, …
automorphism group of	C₂²×D₄
holomorph of	C₂²×C₄

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.4);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.4*F.5*F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);