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#72228688

order11664
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(3,𝔽3)2 ⋊ C2×D4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#13091502C23
Frattini, socleC32#13049813
Fitting, hypocenterH(3,𝔽3)2C2×D4
LCSG ▷ #13091502 ▷ H(3,𝔽3)2
UCSC1

1843 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1843 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 36
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 3 generators on 36 points:

(1,11,9)(2,8,15)(3,4,10)(5,18,13)(6,7,16)(12,14,17)(19,29)(20,32)(21,30)(23,24)(25,34)(26,36);(3,9)(4,5)(6,12)(7,15)(8,16)(10,11)(14,17)(21,26)(22,29)(23,32)(24,25)(27,36)(28,33)(30,31);(1,2)(3,8,10,15)(4,12)(5,6)(7,11,16,9)(13,14,18,17)(19,20)(21,25)(22,28,30,32)(23,31,33,29)(24,26)(27,35,36,34)

Polycyclic presentation:

2^4,3^6|a^2=d,b^a=bd,e^a=eg,e^b=egj,e^c=e^2j^2,e^d=e^2j^2,f^a=fh,f^b=fhi,f^c=efj^2,f^d=f^2i^2,g^a=eg^2j,g^b=g^2,g^c=g^2j,g^d=g^2j,g^e=gj,g^f=gj,h^a=fh^2i,h^b=h^2,h^c=ghj,h^d=h^2i,h^e=hj,h^f=hi,i^b=i^2,j^b=j^2

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

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