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#72228486

order7776
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ #127

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×C3⋊D32C23
Fitting, socle, hypocenterC35#127
LCSG ▷ C3×C3⋊D32 ▷ C35
UCSC1

1592 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 1592 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 15
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2,3)(4,5,9,8)(6,7)(10,15)(11,12,14,13);(2,3)(7,8)(12,13);(10,11)(12,14,13,15)

Polycyclic presentation:

2^5,3^5|a^2=d,c^2=e,b^a=bd,c^b=ce,f^b=f^2,g^a=gh,g^b=g^2h^2,g^c=g^2h^2,g^d=g^2,g^e=g^2,h^a=gh^2,h^c=g^2h,h^d=h^2,h^e=h^2,i^a=gi,i^b=i^2,i^c=i^2j^2,i^d=g^2hi,i^e=i^2,j^a=hj,j^b=i^2j,j^c=i^2j,j^d=gj,j^e=j^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259512, D3×#72213247
maximal normal subgroup of#72259512, D3×#72213247

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