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#72242853

order1536
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

derived#2679⋊C3 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC22#85559
commutator, hypocenter#2679⋊C3C4
Frattini, socleC24#1020
FittingC2×#2679D3
prime radical, cosocleC2×#2679⋊C3C2
LCSG ▷ #2679⋊C3
UCSC1 ◁ C22

1303 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1303 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 16
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^2,3,2^7|a^2=b,c^a=c^2hi,c^b=chi,d^a=e,d^c=e,e^a=d,e^c=de,f^a=fg,f^c=g,f^d=fh,f^e=fhi,g^c=fg,g^d=gj,g^e=gi,h^c=i,i^a=j,i^c=j,j^a=i,j^c=h

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#11714279, #72261991
maximal normal subgroup ofC4×#11714279
maximal quotient of#72242927

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