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#72226353

order31104
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC24×C33 ⋊ C2×D32
derived#72220499 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72220499C23
FrattiniC32D3×S42
FittingC24×C33C2×D32
socleC24×C32D33
LCSG ▷ #72220499
UCSC1

At least 6593 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has at least 6593 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 26 points:

(1,3,2)(5,7,8)(9,10,15,14,11,13)(12,16)(18,24,20,19,21,23)(25,26);(3,4)(6,7)(10,12)(13,17)(14,16)(19,22)(23,26)(24,25);(6,7)(20,21)(23,24)(25,26)

Polycyclic presentation:

2^3,3^5,2^4|d^b=d^2,d^c=d^2,e^a=e^2,e^b=e^2,e^d=eg,f^a=e^2f,f^b=f^2,f^d=fgh^2,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=j,i^b=j,i^e=ij,i^f=j,j^a=i,j^b=i,j^e=i,j^f=ij,k^a=ijk,k^b=ijk,k^e=ijk,k^f=l,l^b=jkl,l^e=il,l^f=kl

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

holomorph ofC3⋊S4

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