#72268140

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#11715569
commutator	C₂×#543⋊C₃	C₂×C₄
Frattini	C₂²×C₄	C₂×#1020
Fitting	C₂×#62844	D₃
prime radical	C₂×#62844⋊C₃	C₂
cosocle	C₂×#3459⋊C₃	C₂²
hypercenter	C₂×C₂²⋊C₄	#1020
hypocenter	#543⋊C₃	C₂²⋊C₄
LCS	G ▷ C₂×#543⋊C₃ ▷ #543⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₄

7577 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 7577 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 64

Polycyclic presentation:

2^4,3,2^6|b^2=c,c^2=jk,h^2=jk,i^2=jk,b^a=bd,c^a=cj,d^b=dj,e^b=e^2,f^b=fg,f^e=g,g^e=fg,h^b=hi,h^e=i,h^g=hj,i^b=ijk,i^e=hijk,i^f=ij,i^h=ijk

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268143, #72268161, #72271326, …
maximal normal subgroup of	#72268143, #72268161, #72271326, …

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