Finite Group Databasebeta

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#72258501

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#4928925 ⋊ D3

Subgroups

subgroupG/H
centerC22#11713204
commutator, cosocleC2×#376⋊C3C23
FrattiniC23×C4C22×S4
Fitting#4928925D3
socleC24C2×GL(2,ℤ/4)
hypercenterC4⋊C4#857⋊C2
hypocenter#95⋊C3#120
LCSG ▷ C2×#376⋊C3 ▷ #95⋊C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

4624 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 4624 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4)(5,20,27,6,18,26)(7,13,28,10,9,25)(8,19,23,12,15,17)(11,16,24,14,21,22);(2,4)(6,12)(10,11)(13,21)(19,20)(22,28)(23,27);(6,11)(7,14)(9,17)(10,12)(13,22)(15,24)(16,26)(18,25)(19,27)(20,23)(21,28)

Polycyclic presentation:

2^5,3,2^5|c^2=ek,d^2=k,g^2=i,h^2=j,c^a=cd,c^b=ce,d^a=dk,d^c=dk,f^a=f^2,f^c=fjk,f^d=fik,g^a=hijk,g^b=gik,g^f=gh,h^a=gijk,h^b=hjk,h^f=gjk,h^g=hk,i^a=j,i^f=ijk,j^a=i,j^f=i

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72258536

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