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#72229294

order2592
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derived(C22⋊A4)⋊C27 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter(C22⋊A4)⋊C27C2
FrattiniC9A4⋊S4
FittingC24×C9C3⋊D3
socleC24×C3C3⋊D9
LCSG ▷ (C22⋊A4)⋊C27
UCSC1

233 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 35
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 35 points:

(2,5)(3,9)(4,12)(6,14)(7,17)(8,20)(10,22)(11,23)(13,21)(15,24)(16,26)(18,27)(19,25)(28,31,30,29)(32,35,33,34);(1,7,13,11,26,27,9,2,15,4,19,20,22,14,6,10,8,25,12,24,5,3,18,16,23,21,17)(33,34,35);(29,30,31)(33,34,35)

Polycyclic presentation:

2,3^4,2^4|c^3=d,d^3=e,b^a=b^2,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^b=g,f^c=g,g^a=fg,g^b=fg,g^c=fg,h^a=ghi,h^b=i,h^c=fgh,i^a=fi,i^b=hi,i^c=fi

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72229343, C2×#72229294
maximal normal subgroup of#72229343, C2×#72229294
maximal quotient ofC2×#72229294, #72276227

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