Finite Group Databasebeta

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#72228079

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×H(7,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×#1020
commutator, cosocleC2×#997C23
FittingC2×H(7,𝔽2)D3
hypercenterC2×D4#1020
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×D4

7467 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 7467 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 20
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^4,3,2^5|b^2=d,f^2=j,g^2=j,h^2=j,i^2=j,b^a=bd,c^a=cj,c^b=cj,e^b=e^2,f^b=fg,f^e=g,g^b=gj,g^e=fgj,g^f=gj,h^b=hi,h^e=i,i^b=ij,i^e=hij,i^h=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228081, #72228096, #72228367, …
maximal normal subgroup of#72258828

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