Finite Group Databasebeta

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#72224532

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, SylowC35 ⋊ C2×C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC35C2×C8
prime radical#11758168C4
cosocleC3×#66472C22
LCSG ▷ C35
UCSC1

468 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 21
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 21 points:

(2,3)(4,12,10,8,5,6,11,7)(13,16,19,21,17,14,20,15);(1,2,3)(5,7,8,6,9,10,12,11)(14,17,19,21,20,18,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224533, D3×#13049788
maximal normal subgroup of#72224533, D3×#13049788

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