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#72228290

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
hypercenterC8A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

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

normal subgroups only

This group has 650 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)(2,11)(4,64,32,68)(5,29,8,6)(7,30)(10,17)(12,81,50,85)(13,39,16,35)(14,34,48,37)(15,40)(19,92,60,93)(20,21,23,52)(24,27,61,25)(26,67,28,62)(31,63,65,66)(33,86,73,87)(36,47)(38,80,70,84)(41,45,77,43)(42,71,79,69)(44,96,46,78)(49,75,82,83)(51,90,57,94)(54,58,89,56)(55,88,91,59)(72,74,76,95);(1,12,55,11,28,74,18,33,24,2,19,75,9,46,54,10,4,42,3,38,63,17,51,41)(5,44,93,37,32,87,23,70,67,13,57,85,8,50,94,34,26,84,20,73,68,16,60,96)(6,45,91,39,65,95,52,71,61,14,58,83,29,82,89,35,27,79,21,76,66,48,88,77)(7,49,90,40,25,80,53,72,64,15,59,78,30,43,92,36,31,86,22,69,62,47,56,81);(5,6,30)(7,8,29)(13,14,47)(15,16,48)(20,21,53)(22,23,52)(25,26,27)(31,32,65)(34,35,40)(36,37,39)(43,44,45)(49,50,82)(56,57,58)(59,60,88)(61,62,67)(64,68,66)(69,70,71)(72,73,76)(77,78,96)(79,80,84)(81,85,83)(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=bc,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#72258953, #72258961
maximal normal subgroup of#72258953, #72258961

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);;
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);