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#72228154

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting#153709 ⋊ D3
derived#72222569 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC2#72209990
commutator, cosocle, hypocenter#72222569C2
Frattini, socleC25#1020
Fitting#153709D3
LCSG ▷ #72222569
UCSC1 ◁ C2

1707 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 1707 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 32
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 32 points:

(1,4,5,7)(9,16,12,11)(10,17,14,18,15,13,20,19)(21,25,24,23)(22,30,27,26,28,32,29,31);(2,3,4)(6,7,8)(9,10,13)(11,14,19)(12,15,17)(16,20,18)(21,22,26)(23,27,32)(24,28,31)(25,29,30)

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=h,e^2=i,f^2=j,b^a=b^2,c^a=dg,c^b=dgk,d^a=ch,d^b=cdk,d^c=dk,e^a=cfhij,e^b=ef,e^c=ek,e^d=ek,f^a=degij,f^b=ej,f^c=fk,g^a=h,g^b=h,h^a=g,h^b=ghk,i^a=gjk,i^b=ij,j^a=hik,j^b=i

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228157, C2×#72228154, #72239184, …
maximal normal subgroup ofC2×#72228154, #72239184, #72239370, …
maximal quotient ofC2×#72228154

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", "k");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.4*F.7);;
gap> SetConjugate(coll, 3, 2, F.4*F.7*F.11);;
gap> SetConjugate(coll, 4, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.3*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.4*F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);