Finite Group Databasebeta

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#72260373

order28800
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, socleC24×C52 ⋊ D3×Dic3
derivedC52×A42 ⋊ C2×C4
SylowC52 ⋊ S4×SL(2,ℤ/4)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterC52×A42C2×C4
Fitting, socleC24×C52D3×Dic3
prime radicalA4×(C5⋊S4)⋊D5C2
cosocleA42×C5⋊D5C22
LCSG ▷ C52×A42
UCSC1

2674 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 2674 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 18
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 18 points:

(1,4,2,3)(5,7,8,6)(10,11,12,13)(14,15,16,17);(2,3,4)(7,8)(9,10,13,11,12)(15,16)(17,18)

Polycyclic presentation:

2^3,3^2,5^2,2^4|b^2=c,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^4,f^b=f^3,f^c=f^4,g^a=f^3g,g^b=g^3,g^c=g^4,h^d=i,h^e=hi,i^a=hi,i^b=hi,i^d=hi,i^e=h,j^a=hj,j^b=hj,j^d=hj,j^e=k,k^b=ijk,k^d=hik,k^e=jk

Conjugacy classes & character table

122 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 5, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^3*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> SetConjugate(coll, 10, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.8*F.10);;
gap> SetConjugate(coll, 10, 4, F.8*F.10);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.8*F.9*F.11);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);