Finite Group Databasebeta

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#72276160

order7776
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

SylowH(5,𝔽3) ⋊ C4⋊Q8

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC2×C3#13049818
commutator, prime radical, cosocleC2×#92404C23
FittingC2×H(5,𝔽3)C2×Q8
hypocenterH(5,𝔽3)C4⋊Q8
LCSG ▷ C2×#92404 ▷ H(5,𝔽3)
UCSC1 ◁ C2×C3

578 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 108

Polycyclic presentation:

2^5,3^5|a^2=e,b^2=d,c^2=e,b^a=bd,c^a=ce,f^a=fg,f^b=f^2,f^c=g^2j^2,f^e=f^2,g^a=fg^2j^2,g^b=g^2j,g^c=fj^2,g^e=g^2j,g^f=gj,h^a=hi,h^b=ghj,h^c=h^2i,h^e=h^2,h^f=hj,i^a=hi^2,i^b=fgi,i^c=hi,i^e=i^2,i^f=ij,i^g=ij^2

Conjugacy classes & character table

58 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, 2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.7^2*F.10^2);;
gap> SetConjugate(coll, 6, 5, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6*F.7^2*F.10^2);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.10^2);;
gap> SetConjugate(coll, 7, 5, F.7^2*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);