Finite Group Databasebeta

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#72240895

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

Aliases

No aliases for this group.

Splittings

Fitting#8698894 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#11715753
commutator, cosocle, hypocenter#2679⋊C3C23
Frattini, socleC23C22×#1020
Fitting#8698894D3
hypercenterC4#11715753
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C4

5410 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 5410 conjugacy classes of subgroups — too many to draw. Only its 45 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240903, C2×#72240895
maximal normal subgroup of#72240903, C2×#72240895
maximal quotient ofC2×#72240895

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, 1, 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.4^2);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.5);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.8*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, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*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, 2, 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, 2, F.10*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);