Finite Group Databasebeta

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#72222012

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

Aliases

No aliases for this group.

Splittings

Fitting, SylowC32⋊H(3,𝔽3) ⋊ C2×SD16

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#66472⋊C3C23
Frattini, socleC32D3×AΓL(1,𝔽9)
Fitting, hypocenterC32⋊H(3,𝔽3)C2×SD16
prime radical#72221995C2
LCSG ▷ #66472⋊C3 ▷ #92301 ▷ C32⋊H(3,𝔽3)
UCSC1

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

normal subgroups only

This group has 473 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

Polycyclic presentation:

2^5,3^5|c^2=e,d^2=e,c^b=cd,d^b=de,d^c=de,f^a=f^2,f^b=f^2,g^a=g^2,g^b=g^2h^2,g^c=g^2h,g^d=gh,g^e=g^2,g^f=gi,h^a=h^2,h^c=gh,h^d=gh^2,h^e=h^2,h^f=hi^2j,i^b=j,i^c=ij,i^d=j,i^e=i^2,j^b=i,j^c=ij^2,j^d=i^2,j^e=j^2

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72222023

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