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#72240893

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#59816 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#11715754
commutator, cosocle, hypocenter#2679⋊C3C22
Frattini, socleC23C2×#1020
Fitting#59816D3
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2

1142 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 12
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

2^2,3,2^7|a^2=hij,f^2=h,g^2=hj,c^a=c^2,c^b=c^2h,d^a=e,d^b=ehj,d^c=e,e^a=d,e^b=dh,e^c=de,f^a=gh,f^b=gh,f^c=gh,f^d=fh,f^e=fhi,g^a=fhj,g^b=fhj,g^c=fg,g^d=ghi,g^e=ghj,h^a=hj,h^b=hj,h^c=hj,i^a=ij,i^b=ij,i^c=hi,j^c=h

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240898, #72240902, C2×#72240893
maximal normal subgroup of#72240898, #72240902, C2×#72240893
maximal quotient ofC2×#72240893

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