Finite Group Databasebeta

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#72341524

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

Aliases

No aliases for this group.

Splittings

FittingQ8×#120 ⋊ D3

Subgroups

subgroupG/H
centerC2C22×#1020⋊C2
commutator, cosocleC2×C22⋊A4C24
Frattini, socleC23C23×S4
FittingQ8×#120D3
hypercenterQ8#1020⋊C2
hypocenterC22⋊A4C2×#41
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2 ◁ Q8

3601 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 3601 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#41×#1020⋊C2
maximal normal subgroup of#41×#1020⋊C2
maximal quotient of#72341613

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