Finite Group Databasebeta

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#72276569

order7776
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derivedC22⋊C92 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊C92C2×C3
FrattiniC32C3×A4⋊S4
FittingC24×C33C3⋊D3
socleC24×C32C3×C3⋊D3
LCSG ▷ C22⋊C92
UCSC1

940 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 26
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 3 generators on 26 points:

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

Polycyclic presentation:

2,3^5,2^4|c^3=e,d^3=f,c^a=c^2e^2,c^b=ce^2,d^a=d^2f^2,d^b=de,e^a=e^2,f^a=f^2,g^c=h,g^d=gh,h^a=gh,h^c=gh,h^d=g,i^a=gi,i^c=gi,i^d=j,j^a=hij,j^c=ghj,j^d=ij

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

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