Finite Group Databasebeta

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#72220466

order10368
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC34 ⋊ #1629

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle#1572C23
Fitting, socle, hypocenterC34#1629
prime radicalGO+(4,𝔽2)2C2
LCSG ▷ #1572 ▷ C3⋊D32 ▷ C34
UCSC1

1385 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1385 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,3^4|b^2=f,d^2=f,e^2=g,b^a=be,c^a=cg,c^b=cd,d^a=dg,d^b=df,d^c=df,e^a=eg,e^b=eg,e^c=eg,h^a=h^2,h^b=i,h^c=hi^2,h^d=hi^2,h^e=hi^2,h^f=h^2,h^g=h^2,i^a=hi,i^b=h^2,i^c=i^2,i^d=h^2i^2,i^e=h^2i^2,i^f=i^2,i^g=i^2,j^b=hjk^2,j^c=h^2jk,j^d=hj^2k^2,j^e=hij,j^f=j^2,j^g=hi^2j,k^a=h^2k,k^b=hij^2k^2,k^c=h^2i^2k^2,k^d=i^2j^2k,k^e=h^2k,k^f=k^2,k^g=hik

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAΓL(1,𝔽9)2
maximal normal subgroup ofAΓL(1,𝔽9)2
holomorph ofGO+(4,𝔽2)

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