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#72225982

order2304
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22S42
commutator, cosocleC2×Ω+(4,𝔽3)C22
FittingC2×H(5,𝔽2)D32
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22

680 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 52
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 52 points:

(2,4)(5,49,26,27)(6,33,28,30)(7,8,20,9)(10,29,11,23)(12,21,25,22)(13,37,17,14)(15,34,31,51)(16,35,24,36)(18,40,32,39)(19,43)(41,46,45,48)(42,50);(1,2,3,4)(5,18,19,6,7,14)(8,44,41,24,20,37)(9,47,42,17,21,51)(10,38,45,15,29,33)(11,52,50,28,49,35)(12,32,43,13,23,30)(16,27,34,26,39,48)(22,36,25,40,46,31)

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,f^2=j,g^2=j,b^a=bcj,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=fj,f^b=fj,f^d=fg,f^e=g,g^a=fgj,g^b=fgj,g^d=f,g^e=fg,g^f=gj,h^a=fh,h^b=fgi,h^d=fhj,h^e=i,h^f=hj,h^g=hj,i^a=ij,i^b=gh,i^d=gij,i^e=hi,i^g=ij

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226236, #72226254, #72226288
maximal normal subgroup of#72226236, #72226254, #72226288

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