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#72245595

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC4×B(4,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocleC2×#997C23
FrattiniC24C22×S4
FittingC4×B(4,𝔽2)D3
hypercenterC2×C4#1020⋊C2
hypocenter#997#41
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4

987 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 987 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6)(3,8)(9,20,18)(10,19,21)(11,22,13)(12,17)(14,24,16)(15,23);(1,3)(2,5)(4,8)(6,7)(10,14)(12,20)(13,18)(15,24)(17,22)(19,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19)(13,21)(15,20)(16,18)(17,24)(22,23)

Polycyclic presentation:

2^4,3,2^5|f^2=j,g^2=j,b^a=bd,c^a=cdj,c^b=cdj,e^a=e^2,e^b=e^2,f^a=fg,f^b=fg,f^c=fhj,f^e=g,g^a=gj,g^b=gj,g^c=ghij,g^e=fgj,g^f=gj,h^a=i,h^b=i,h^e=hi,h^g=hj,i^a=h,i^b=h,i^e=h,i^f=ij,i^g=ij

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72246216, #72246231, #72246847, …
maximal normal subgroup of#72246216, #72246231, #72246847, …
maximal quotient ofC2×#72245595

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