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#72258077

order2304
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25×C3

Aliases

No aliases for this group.

Splittings

FittingC22×C3×H(5,𝔽2) ⋊ D3
derivedC3×C22⋊#997 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊#87588
commutator, cosocle, hypocenterC3×C22⋊#997C2
FittingC22×C3×H(5,𝔽2)D3
socleC23×C3#1020
LCSG ▷ C3×C22⋊#997
UCSC1 ◁ C2

2192 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 2192 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 72

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

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

Polycyclic presentation:

2,3^2,2^7|d^2=j,e^2=j,f^2=j,g^2=j,b^a=b^2,c^a=c^2,d^a=ej,d^b=e,e^a=dj,e^b=dej,e^d=ej,f^a=dfg,f^b=g,f^e=fj,g^a=deg,g^b=fgj,g^d=gj,g^f=gj,h^a=i,h^b=i,h^e=hj,h^g=hj,i^a=h,i^b=hi,i^d=ij,i^f=ij

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72258083

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