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#72227573

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC23×2−1+4 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#87524
commutator, cosocle#5948C23
FrattiniC4C22×#1020
FittingC23×2−1+4D3
socleC23C2×GL(2,ℤ/4)
hypercenterC2×D4#1020
hypocenterC22⋊SL(2,𝔽3)C2×D4
LCSG ▷ #5948 ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

7099 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 7099 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 96

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

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

Polycyclic presentation:

2^4,3,2^5|c^2=j,d^2=j,h^2=j,i^2=j,c^a=cd,c^b=cj,d^a=dj,d^c=dj,e^c=e^2,f^c=fg,f^e=g,g^e=fg,h^c=fij,h^e=i,i^c=fghj,i^e=hij,i^h=ij

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72227574, #72227580, #72267944

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