Finite Group Databasebeta

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#72341141

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

Aliases

No aliases for this group.

Splittings

Fitting#8075294 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72218857
commutator, cosocleC2×#5464C23
Frattini, socleC23C22×#1020
Fitting#8075294D3
hypocenter#5464C2×D4
LCSG ▷ C2×#5464 ▷ #5464
UCSC1 ◁ C2

5836 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 5836 conjugacy classes of subgroups — too many to draw. Only its 51 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:

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

Polycyclic presentation:

2^4,3,2^6|b^2=d,h^2=k,i^2=jk,b^a=bd,e^c=e^2,f^a=fj,f^b=fj,f^c=fgjk,f^e=g,g^a=gk,g^b=gk,g^c=gk,g^e=fg,h^a=hj,h^b=hk,h^c=fi,h^e=ik,h^f=hk,h^g=hj,i^a=ik,i^b=ijk,i^c=fghjk,i^e=hi,i^f=ij,i^g=ijk,j^c=jk,j^e=k,k^e=jk

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341172, C2×#72341141
maximal normal subgroup of#72341172, C2×#72341141
maximal quotient ofC2×#72341141

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