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#72253122

order5184
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

Fitting3−1+2×2−1+4 ⋊ D3

Subgroups

subgroupG/H
centerC2C22×A4⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9×SL(2,𝔽3)C23×C3
Frattini, socleC2×C3C22×C3×C3⋊S4
Fitting3−1+2×2−1+4D3
hypercenterD4A4⋊AGL(1,ℤ/9)
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ D4

1393 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1393 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

2^3,3^4,2^3|a^2=j,f^3=g,h^2=j,i^2=j,c^a=cj,e^b=e^2,f^b=f^2g^2,f^d=fg,g^b=g^2,h^b=ij,h^e=hi,h^f=hi,i^b=hj,i^e=h,i^f=h,i^h=ij

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268997

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