#72246783

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#72238589 ⋊ D₃
derived	#11710730 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, socle	C₂²	#72209653
commutator, hypocenter	#11710730	C₂×C₄
Frattini	C₂⁵	C₂×#1020
Fitting	#72238589	D₃
prime radical	C₂×#72209772	C₂
cosocle	C₂×#11710730	C₂²
hypercenter	C₂³	#87588⋊C₂
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂² ◁ C₂³

At least 7677 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has at least 7677 conjugacy classes of subgroups — too many to draw. Only its 40 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:

2^3,3,2^8|a^2=c,b^2=c,d^a=d^2,e^a=f,e^b=eij,e^d=f,f^a=e,f^b=fi,f^d=ef,g^a=h,g^d=h,g^f=gk,h^a=g,h^d=gh,h^e=hk,i^a=ij,i^d=j,i^g=il,j^d=ij,j^g=jl,j^h=jl,k^b=kl

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253698, C₄×#72209832
maximal normal subgroup of	C₄×#72209832

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);