Finite Group Databasebeta

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#72341297

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#21481 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC24C22×S4
Fitting#21481D3
prime radicalC2×#5935C22
cosocleC2×#997C23
hypercenterC2×C4#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

1090 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(5,14,16,8,9,7,18,15)(6,13,19,12,10,11,20,17);(1,2,3,4)(6,10)(7,15,18,14,8,16)(11,12,20)(13,17,19);(5,6,9,10)(7,13,14,11)(8,17,15,12)(16,20,18,19)

Polycyclic presentation:

2^4,3,2^5|a^2=j,b^2=d,f^2=j,g^2=j,b^a=bj,e^c=e^2,f^a=fij,f^b=fij,f^c=gj,f^e=g,g^a=ghj,g^b=ghj,g^c=fj,g^e=fgj,g^f=gj,h^c=i,h^e=hi,h^f=hj,i^c=h,i^e=h,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341309, C4×#11714182, #72341419
maximal normal subgroup of#72341309, C4×#11714182, #72341419

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