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#72224533

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ C2×SD16

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×#66472C23
Fitting, socle, hypocenterC35C2×SD16
prime radical#72224531C2
LCSG ▷ C3×#66472 ▷ C3×C3⋊(C3⋊(C3⋊D3)) ▷ C35
UCSC1

862 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 21
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224537, D3×#72210346
maximal normal subgroup ofD3×#72210346

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