Finite Group Databasebeta

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#72226067

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

Aliases

No aliases for this group.

Splittings

Fitting#14713 ⋊ D3

Subgroups

subgroupG/H
centerC22C2×#857⋊C2
commutatorC2×C42⋊C3C22×C4
Frattini, socleC24C22×S4
Fitting#14713D3
prime radicalC22×#857⋊C2C2
cosocleC22×C42⋊C3C23
hypercenterC2×C4#857⋊C2
hypocenterC42⋊C3C2×C22⋊C4
LCSG ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C22 ◁ C2×C4

2441 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 2441 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 16 points:

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

Polycyclic presentation:

2^5,3,2^4|c^2=e,g^2=i,h^2=j,c^b=cd,f^a=f^2,f^b=f^2,g^a=hij,g^b=hi,g^f=gh,h^a=gij,h^b=gj,h^f=gj,i^a=j,i^b=j,i^f=ij,j^a=i,j^b=i,j^f=i

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258638
maximal normal subgroup of#72258638
maximal quotient of#72226072, #72228324

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