#72209653

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	#35346 ⋊ D₃
derived	C₂²⋊#997 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#87588⋊C₂
commutator, cosocle, hypocenter	C₂²⋊#997	C₂²
Frattini, socle	C₂³	C₂×#1020
Fitting	#35346	D₃
LCS	G ▷ C₂²⋊#997
UCS	C₁ ◁ C₂

2511 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 2511 conjugacy classes of subgroups — too many to draw. Only its 22 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:

2^2,3,2^7|f^2=j,g^2=j,c^b=c^2,d^b=de,d^c=e,e^c=de,f^a=fi,f^b=defghij,f^c=g,f^e=fj,g^a=gh,g^b=egh,g^c=fgj,g^d=gj,g^f=gj,h^c=hi,i^b=hi,i^c=h

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209654, #72209774, #72217073, …
maximal normal subgroup of	#72209774, #72217073, #72217499, …
maximal quotient of	#72209832, #72209837, #72209838, …
outer automorphism group of	C₂²×D₈, C₂²×Q₁₆, C₂²×Dic₆, …

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