Finite Group Databasebeta

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#72228540

order3888
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ C2×D4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC33×C3⋊D3C23
Fitting, socle, hypocenterC35C2×D4
LCSG ▷ C33×C3⋊D3 ▷ C35
UCSC1

2180 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 2180 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 15
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 15 points:

(2,3)(4,5)(7,8)(10,12)(11,15);(1,2,3)(8,9)(12,13);(5,6)(8,9)(10,11)(12,14,13,15)

Polycyclic presentation:

2^4,3^5|a^2=d,b^a=bd,e^c=e^2,f^a=fg,f^b=f^2g^2,f^c=f^2,f^d=f^2,g^a=fg^2,g^c=g^2,g^d=g^2,h^a=f^2g^2h^2,h^b=h^2,h^c=h^2,h^d=f^2gh,i^a=g^2i^2,i^b=hi,i^c=i^2,i^d=f^2g^2i

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×(#157⋊D3)⋊D3
maximal normal subgroup ofD3×(#157⋊D3)⋊D3

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