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#72209952

order5832
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C32

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽3) ⋊ SL(2,𝔽3)
derived#14519166 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC3#14519481
commutator, cosocle, hypocenter#14519166C3
FittingH(5,𝔽3)SL(2,𝔽3)
LCSG ▷ #14519166
UCSC1 ◁ C3

309 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 10 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:

3,2^3,3^5|b^2=d,c^2=d,b^a=c,c^a=bcd,c^b=cd,e^a=ef^2g^2h,e^b=e^2f^2gi^2,e^c=ef^2gh^2i^2,e^d=e^2i^2,f^a=fh,f^b=e^2fgh^2i^2,f^c=e^2f^2h^2i,f^d=f^2i^2,g^a=gh^2,g^b=g^2h^2i,g^c=gh^2,g^d=g^2i,g^f=gi,h^b=g^2hi^2,h^c=g^2h^2i^2,h^d=h^2,h^e=hi^2

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72254100
maximal subgroup of#72209953, #72210405, #72214508, …
maximal normal subgroup of#72210405, #72214508, #72214540, …
maximal quotient ofC2×#72209952

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