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#72254100

order23328
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽3) ⋊ C2×GL(2,𝔽3)
derived#72209952 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72209952C22
Frattini, socleC3#72210382
FittingH(5,𝔽3)C2×GL(2,𝔽3)
LCSG ▷ #72209952
UCSC1

953 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2^2,3,2^3,3^5|d^2=f,e^2=f,c^b=c^2,d^b=df,d^c=e,e^b=de,e^c=def,e^d=ef,g^a=ghk^2,g^b=gh,g^d=hj^2,g^e=ghj^2k^2,g^f=g^2k,h^a=h^2,h^b=h^2,h^d=i,h^e=hi,h^f=h^2,i^a=i^2,i^c=hi,i^d=h^2,i^e=hi^2,i^f=i^2,i^g=ik,j^a=hi^2j,j^b=ij^2k,j^c=g^2j,j^d=gik^2,j^e=g^2h^2ij^2k,j^f=j^2k,j^g=jk,j^h=jk^2,k^a=k^2

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

automorphism group ofC3⋊#272

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