Finite Group Databasebeta

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#72240899

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#8934366 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72218857
commutator, cosocle, hypocenter#2679⋊C3C23
Frattini, socleC23C22×#1020
Fitting#8934366D3
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2

4851 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 4851 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^3,3,2^7|c^2=ijk,g^2=i,h^2=ik,c^b=cijk,d^a=dik,d^b=dik,d^c=d^2,e^a=ei,e^c=f,e^d=f,f^a=fik,f^c=e,f^d=ef,g^b=gi,g^c=hi,g^d=hi,g^e=gi,g^f=gij,h^b=hik,h^c=gik,h^d=gh,h^e=hij,h^f=hik,i^c=ik,i^d=ik,j^c=jk,j^d=ij,k^d=i

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240903, #72340959, C2×#72240899
maximal normal subgroup of#72240903, C2×#72240899
maximal quotient ofC2×#72240899

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