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#72228292

order2304
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#11923523
commutator, cosocle(#41⋊C3)⋊A4C22
FrattiniC4C2×A4⋊S4
FittingH(7,𝔽2)C3⋊D3
prime radical(2−1+4⋊C3)⋊A4C2
hypercenterD4A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

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

normal subgroups only

This group has 831 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) = 96
transitive degreeμt(G) = 96

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

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,c^2=j,f^2=j,g^2=j,b^a=bcj,c^b=cj,d^a=d^2,e^a=e^2,f^a=gj,f^d=g,f^e=g,g^a=fj,g^d=fgj,g^e=fgj,g^f=gj,h^a=fhi,h^d=i,h^e=fgh,h^g=hj,i^a=fgij,i^d=hi,i^e=fij,i^f=ij

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258941
maximal normal subgroup of#72258941

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