Finite Group Databasebeta

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#72214583

order3888
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowH(5,𝔽3) ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterH(5,𝔽3)C2×C8
Frattini, socleC3#13049788
prime radical#11758087C4
cosocle#11758052C22
LCSG ▷ H(5,𝔽3)
UCSC1

210 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3^5|b^2=c,c^2=d,e^a=e^2,e^b=eg,e^c=e^2gi^2,e^d=e^2,f^a=fgi,f^b=efgh^2i,f^c=ef^2gh^2,f^d=f^2i^2,f^e=fi,g^a=g^2i,g^b=ei,g^c=egi^2,g^d=g^2i,g^e=gi,h^a=g^2hi^2,h^b=e^2f^2gi^2,h^c=e^2f^2gh,h^d=h^2,h^e=hi^2,h^f=hi^2,i^b=i^2

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214604, #72228597, #72228622
maximal normal subgroup of#72214604, #72228597, #72228622
maximal quotient of#72276127

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