Finite Group Databasebeta

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#72236903

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC2×#3027 ⋊ D3

Subgroups

subgroupG/H
center, socleC2C2×#73081
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
FittingC2×#3027D3
hypercenterC2×C4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4

1476 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^4,3,2^5|a^2=d,d^2=j,f^2=j,g^2=j,b^a=bd,c^a=cj,d^b=dj,e^a=e^2,f^a=fg,f^b=fhj,f^e=g,g^a=gj,g^b=ghij,g^e=fgj,g^f=gj,h^a=i,h^e=hi,h^g=hj,i^a=h,i^e=h,i^f=ij,i^g=ij

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240979, #72240988, #72240998, …
maximal normal subgroup of#72240979, #72240988, #72240998, …
maximal quotient of#72341747, C2×#72236903, #72343791

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