Finite Group Databasebeta

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#72235300

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

derived#5464 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenter#5464C2×C4
FrattiniC23C2×#1020
Fitting#63011D3
socleC22C2×#5936
prime radical#63011⋊C3C2
cosocle#3532⋊C3C22
LCSG ▷ #5464
UCSC1

1272 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,18,20,6)(2,32,12,26)(3,8,14,21)(4,23,15,7)(5,16,22,17)(9,29,28,10)(11,31,24,25)(13,27,19,30);(1,9,8,13,22,2)(3,27,23,30,17,19)(4,31,15,32,20,10)(5,26,14,12,6,11)(7,25)(16,29,21,24,18,28)

Polycyclic presentation:

2^3,3,2^6|b^2=c,g^2=ij,h^2=i,b^a=bi,d^a=di,d^b=d^2,e^b=fj,e^c=ej,e^d=f,f^b=ei,f^c=fij,f^d=ef,g^a=gj,g^b=fhij,g^c=gi,g^d=hij,g^e=gj,g^f=gi,h^a=hij,h^b=egj,h^c=hj,h^d=gh,h^e=hi,h^f=hij,i^d=j,j^b=ij,j^d=ij

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341071
maximal normal subgroup of#72341071
maximal quotient of#72235302, #72340948, #72341192

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