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#72226329

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC4S42
commutator, cosocle, hypocenterΩ+(4,𝔽3)C23
Frattini, socleC2C2×S42
Fitting#585D32
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C4

1424 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 96
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 96 points:

(1,25,77,3,43,63,10,62,28,4,26,58)(2,11,15,17)(5,21,74,18,6,96,30,19,81,23,8,76)(7,73,27,20,70,57,9,68,38,22,87,61)(12,86,46,53,52,85,47,82,51,39,45,83)(13,55,14,48,36,54,31,42,50,40,16,41)(24,89,75,56,65,95,67,90,92,44,66,93)(29,37,71,59,34,69,35,33,80,60,32,78)(49,84,79,94)(64,88,72,91);(1,3,10,4)(2,11,15,17)(5,25,30,62)(6,21,8,19)(7,33,9,37)(12,44,47,56)(13,48,31,40)(14,42,50,55)(16,41,36,54)(18,43,23,26)(20,32,22,34)(24,39,67,53)(27,73,38,68)(28,76,77,96)(29,78,35,69)(45,91,52,88)(46,93,51,95)(49,90,79,89)(57,70,61,87)(58,74,63,81)(59,71,60,80)(64,86,72,82)(65,94,66,84)(75,83,92,85);(1,2,10,15)(3,17,4,11)(5,24,30,67)(6,31,8,13)(7,16,9,36)(12,43,47,26)(14,32,50,34)(18,56,23,44)(19,40,21,48)(20,55,22,42)(25,53,62,39)(27,72,38,64)(28,75,77,92)(29,66,35,65)(33,54,37,41)(45,70,52,87)(46,74,51,81)(49,71,79,80)(57,88,61,91)(58,95,63,93)(59,89,60,90)(68,82,73,86)(69,84,78,94)(76,85,96,83)

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,b^2=j,c^2=j,f^2=j,g^2=j,b^a=bj,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

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228062
maximal normal subgroup of#72228062

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.10);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*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);