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#72275397

order3456
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, socleC24×C3 ⋊ GO+(4,𝔽2)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×A4⋊S4C22
Fitting, socleC24×C3GO+(4,𝔽2)
hypocenterC3×A42D4
LCSG ▷ C3×A4⋊S4 ▷ C3×A42
UCSC1

478 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Polycyclic presentation:

2^3,3^3,2^4|b^a=bc,d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2f^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^a=h,g^b=gi,g^c=gi,g^e=h,g^f=hij,h^a=g,h^c=ghj,h^e=gh,h^f=ghj,i^a=gj,i^e=ij,j^a=hi,j^b=ij,j^c=ij,j^e=i

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#10596770⋊C2, C2×#72275397
maximal normal subgroup ofD3×#10596770⋊C2, C2×#72275397
maximal quotient ofC2×#72275397

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