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#72228289

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

Aliases

No aliases for this group.

Splittings

Fitting#3573 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, socleC2C4⋊(A4⋊S4)
commutator, cosocle(#41⋊C3)⋊A4C22
FrattiniC4C2×A4⋊S4
Fitting#3573C3⋊D3
prime radical#11923521C2
hypercenterC8A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

normal subgroups only

This group has 633 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:

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

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^b=d^2,e^b=e^2,f^b=gj,f^d=g,f^e=g,g^b=fj,g^d=fgj,g^e=fgj,g^f=gj,h^b=fhi,h^d=i,h^e=fgh,h^g=hj,i^b=fgij,i^d=hi,i^e=fij,i^f=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258941
maximal normal subgroup of#72258941
maximal quotient of#72253130

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, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, 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, 2, 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, 2, 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);