Finite Group Databasebeta

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#72341564

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC22C2×#1020⋊C2
commutatorC2×C22⋊A4C22×C4
Frattini, socleC24C22×S4
FittingC2×C4×#120D3
prime radicalC22×#120⋊C3C22
cosocleC22×C22⋊A4C23
hypercenterC2×C4#1020⋊C2
hypocenterC22⋊A4C2×C4⋊C4
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C22 ◁ C2×C4

2152 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 96

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^5,3,2^4|a^2=e,b^2=d,c^2=d,b^a=be,c^a=ce,c^b=ce,f^a=f^2,f^c=f^2,g^a=ghj,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^a=ij,j^c=ij,j^f=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#11713201, C4⋊C4×#1020⋊C2, #72341604
maximal normal subgroup ofC4×#11713201, C4⋊C4×#1020⋊C2, #72341604

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, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*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, 1, F.9*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);