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#72227784

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socle, hypercenterC23#1020⋊C2
commutator, hypocenter#543⋊C3C2×C4
FrattiniC25C2×S4
FittingC2×#2029D3
prime radicalC2×#2029⋊C3C2
cosocleC2×#543⋊C3C22
LCSG ▷ #543⋊C3
UCSC1 ◁ C23

787 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 28
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2^3,3,2^6|b^2=c,e^2=j,f^2=j,b^a=bi,d^b=d^2,e^a=ehi,e^b=fj,e^d=ef,f^a=fghi,f^b=ej,f^d=e,f^e=fj,g^d=gh,g^e=gi,g^f=gi,h^b=gh,h^d=g,h^f=hi

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227789

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