Finite Group Databasebeta

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#72341982

order1536
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#392708 ⋊ C3

Subgroups

subgroupG/H
center, socleC22AΣL(1,𝔽64)
commutator#585C2×C3×C4
FrattiniC23×C4C22×A4
Fitting#392708C3
prime radicalC2×#3027⋊C3C2
cosocleC2×#585C22×C3
hypercenterC4⋊C4#120⋊C3
hypocenterH(5,𝔽2)C3×C22⋊C4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

1413 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1413 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 36
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

(6,12)(7,8)(9,25,26,22,10,23)(13,16)(14,30,32,29,15,35)(17,18,34)(19,20,33)(21,28,31,27,24,36);(1,2,3,4)(5,31,7,35,11,32,8,36)(6,33,12,26,16,34,13,23)(9,15,19,30,22,24,17,28)(10,21,20,27,25,14,18,29)

Polycyclic presentation:

2^2,3,2^7|a^2=d,d^2=j,e^2=j,f^2=j,g^2=j,b^a=be,d^b=dj,e^b=ej,f^a=fi,f^b=fi,f^c=fg,g^a=gh,g^b=gh,g^c=f,g^f=gj,h^c=i,h^f=hj,i^c=hij,i^g=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341987, #72342056, #72342074
maximal normal subgroup of#72341987, #72342056, #72342074

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