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#72236838

order1536
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

FittingC2×#3025 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, cosocle, hypocenter#997C24
FrattiniC23C23×S4
FittingC2×#3025D3
socleC2C22×#1020⋊C2
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

3242 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 3242 conjugacy classes of subgroups — too many to draw. Only its 105 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^4,3,2^5|b^2=j,d^2=j,f^2=j,g^2=j,b^a=bj,c^b=cj,d^b=dj,e^b=e^2,f^a=fhj,f^b=fg,f^e=g,g^a=ghij,g^b=gj,g^e=fgj,g^f=gj,h^b=i,h^e=hi,h^g=hj,i^b=h,i^e=h,i^f=ij,i^g=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240979, #72240983, #72240986, …
maximal normal subgroup of#72240979, #72240983, #72240986, …
maximal quotient ofC4×#11714171, C4×#11714175, #72341785, …

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