Finite Group Databasebeta

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#72266639

order1536
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#612382 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#5464
commutator, cosocle, hypocenter#16297C2×C3
Frattini, socleC24C2×C22⋊A4
Fitting#612382C3
prime radical#16297⋊C3C2
LCSG ▷ #16297
UCSC1 ◁ C2×C4

1300 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 96

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

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

Polycyclic presentation:

2,3,2^8|a^2=i,e^2=j,f^2=j,c^b=cd,d^b=c,e^b=f,e^c=eg,e^d=eh,f^b=efj,f^c=fghi,f^d=fgi,f^e=fj,g^b=h,h^b=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266654, #72266655
maximal quotient of#72266643

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