Finite Group Databasebeta

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#72243733

order3840
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

SylowC5 ⋊ #4626⋊C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutatorC5×#127⋊C3C2×C4
FrattiniC23AGL(1,𝔽5)×S4
FittingC5×#127C4×D3
socleC22×C5SL(2,ℤ/4)⋊C4
prime radical, cosocleD5×#127⋊C3C22
hypocenterC5×C42⋊C3C4⋊C4
LCSG ▷ C5×#127⋊C3 ▷ C5×C42⋊C3
UCSC1

458 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2^4,3,5,2^4|a^2=d,b^2=c,g^2=i,h^2=ij,b^a=bd,e^a=e^2,f^b=f^3,f^c=f^4,g^a=gh,g^b=gj,g^d=gi,g^e=hi,h^a=hj,h^b=hi,h^d=hij,h^e=gh,i^a=j,i^e=ij,j^a=i,j^e=i

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243780, AGL(1,𝔽5)×#4626⋊C2
maximal normal subgroup ofAGL(1,𝔽5)×#4626⋊C2

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