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#72219734

order31104
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC22×C34 ⋊ C2×GL(2,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC2C2×#72210382
commutator, cosocleC2×#14519481C23
FittingC22×C34C2×GL(2,𝔽3)
socleC2×C34C22×GL(2,𝔽3)
hypercenterC22#72210382
hypocenter#14519481C2×D4
LCSG ▷ C2×#14519481 ▷ #14519481
UCSC1 ◁ C2 ◁ C22

4309 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 4309 conjugacy classes of subgroups — too many to draw. Only its 59 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:

2^4,3,2^3,3^4|f^2=h,g^2=h,b^a=bd,e^c=e^2,f^c=gh,f^e=fg,g^c=fh,g^e=f,g^f=gh,i^a=i^2,i^c=j,i^e=j,i^f=ij,i^g=ij^2,i^h=i^2,j^a=j^2,j^c=i,j^e=i^2j^2,j^f=ij^2,j^g=i^2j^2,j^h=j^2,k^a=jk,k^c=kl^2,k^e=kl,k^f=l^2,k^g=kl,k^h=k^2,l^a=i^2jl,l^c=l^2,l^f=k,l^g=kl^2,l^h=l^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

holomorph ofC3⋊Dic3, C2×C3⋊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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 5, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 7, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 3, F.11*F.12^2);;
gap> SetConjugate(coll, 11, 5, F.11*F.12);;
gap> SetConjugate(coll, 11, 6, F.12^2);;
gap> SetConjugate(coll, 11, 7, F.11*F.12);;
gap> SetConjugate(coll, 11, 8, F.11^2);;
gap> SetConjugate(coll, 12, 1, F.9^2*F.10*F.12);;
gap> SetConjugate(coll, 12, 3, F.12^2);;
gap> SetConjugate(coll, 12, 6, F.11);;
gap> SetConjugate(coll, 12, 7, F.11*F.12^2);;
gap> SetConjugate(coll, 12, 8, F.12^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);