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#72277301

order5760
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

FittingC5×H(5,𝔽2) ⋊ C3×Dic3
derivedC5×#994 ⋊ C3×C4
SylowC5 ⋊ #859⋊A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×#994C3×C4
FittingC5×H(5,𝔽2)C3×Dic3
socleC2×C5A4×SL(2,ℤ/4)
prime radicalD5×Ω+(4,𝔽3)C2
cosocleD5×#994C2×C3
LCSG ▷ C5×#994
UCSC1 ◁ C2

404 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 37
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(2,3,4,5)(6,21,9,10,11,22,20,23)(7,36,32,14,16,33,26,31,19,27,24,35,15,37,17,29,12,34,13,25,8,28,30,18);(1,2,5,3,4)(6,12,11,16)(7,19,15,8)(9,25,21,30,10,37,20,31,22,24,23,36)(13,35,27,17,14,33,26,18,28,32,29,34)

Polycyclic presentation:

2,3,2^3,3,5,2^3|a^2=c,d^2=j,e^2=j,h^2=j,i^2=j,d^b=e,e^b=dej,e^d=ej,f^a=f^2,g^a=g^3,g^c=g^4,h^a=hj,h^f=i,i^a=hi,i^f=hij,i^h=ij

Conjugacy classes & character table

83 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, 3, 2, 2, 2, 3, 5, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 2, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);