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#72227890

order2304
automorphism?
homologyAbelianization (H1) = C8 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

derivedΩ+(4,𝔽3) ⋊ C8

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#10596770
commutator, hypocenterΩ+(4,𝔽3)C8
FittingC2×H(5,𝔽2)#157
prime radicalC2×Ω+(4,𝔽3)C4
cosocle#11921471C2
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22

168 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 11 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) = 40
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,29,27,13,36,19,30,39)(10,38,28,14,31,32,26,23)(11,16,34,25,12,17,35,37)(15,20,33,18,24,21,22,40);(10,22,11)(12,25,13)(15,26,20)(16,21,19)(17,34,24)(18,31,39)(23,32,38)(27,30,36)(28,35,29)(33,40,37)

Polycyclic presentation:

2^3,3^2,2^5|a^2=b,b^2=c,f^2=j,g^2=j,h^2=j,i^2=j,d^a=de,d^b=d^2,e^a=de^2,e^b=e^2,f^a=hij,f^b=fg,f^d=g,f^e=g,g^a=h,g^b=gj,g^d=fgj,g^e=fgj,g^f=gj,h^a=gj,h^b=hj,h^d=i,i^a=fj,i^b=hi,i^d=hij,i^h=ij

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227892
maximal normal subgroup of#72227892

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