Finite Group Databasebeta

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#72341518

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

Aliases

No aliases for this group.

Splittings

FittingC2×C4×#120 ⋊ D3

Subgroups

subgroupG/H
centerC4C2×#1020⋊C2
commutator, cosocleC2×C22⋊A4C24
Frattini, socleC23C23×S4
FittingC2×C4×#120D3
hypercenterC2×C4#1020⋊C2
hypocenterC22⋊A4C2×#41
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C4 ◁ C2×C4

3929 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 3929 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 16
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 16 points:

(1,4,3)(2,8,7)(9,11,12,15)(10,14,13,16);(1,2)(3,7)(4,6)(5,8)(10,13)(14,16);(1,2)(3,8)(4,6)(5,7)(9,10,12,13)(11,14,15,16);(1,2)(3,8)(4,6)(5,7)

Polycyclic presentation:

2^5,3,2^4|b^2=e,d^2=e,c^b=ce,f^c=f^2,g^a=gij,g^b=gij,g^c=ghj,g^f=h,h^a=hi,h^b=hi,h^c=hi,h^f=gh,i^f=j,j^c=ij,j^f=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#41×#1020⋊C2
maximal normal subgroup of#41×#1020⋊C2
maximal quotient ofC4×#11713201, C4×#11713272

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