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#72228896

order50000
automorphism?
homologyAbelianization (H1) = C42×C5 · Schur multiplier (H2) = C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#15691114 ⋊ C42
derivedC252 ⋊ C42×C5

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC252C42×C5
Frattini, socleC52C5×AGL(1,𝔽5)2
Fitting#15691114C42
prime radical#72213905C22
cosocleD252C22×C5
LCSG ▷ C252
UCSC1

345 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 50
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 50 points:

(2,12,25,7,8,4,17,23,19,21,6,18,16,15,14,13,5,22,24,20)(3,9,10,11)(26,27,46,50,30,32,47,28,29,37,33,40,43,39,49,38,42,36,35,41)(31,44,48,34);(1,15,5,14,20,9,22,19,25,13,3,12,2,6,18,10,4,16,24,23,11,21,8,17,7)(26,30,43,29)(27,44,35,40)(28,37,34,46)(31,47,50,36)(32,45,33,41)(38,39,48,49)

Polycyclic presentation:

2^2,5,2^2,5^4|a^2=d,b^2=e,f^5=h,g^5=i,f^a=f^2h,f^b=f^3h^3,f^c=fh^4,f^d=f^4h^4,f^e=f^4h^4,g^a=fgh,g^b=g^3i^3,g^c=gi^4,g^d=f^3gh^4,g^e=g^4i^4,h^a=h^2,h^b=h^3,h^d=h^4,h^e=h^4,i^a=hi,i^b=i^3,i^d=h^3i,i^e=i^4

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

holomorph of#1035

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