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#72228294

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

Aliases

No aliases for this group.

Splittings

Fitting2−1+6 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, socleC2#11923523
commutator, cosocle(#41⋊C3)⋊A4C22
FrattiniC4C2×A4⋊S4
Fitting2−1+6C3⋊D3
hypercenterQ8A4⋊S4
hypocenterΩ+(4,𝔽3)D4
LCSG ▷ (#41⋊C3)⋊A4 ▷ Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

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

normal subgroups only

This group has 858 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:

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

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,d^b=d^2,e^a=e^2,e^b=e^2,f^a=gj,f^b=gj,f^d=g,f^e=g,g^a=fj,g^b=fj,g^d=fgj,g^e=fgj,g^f=gj,h^a=fhi,h^b=fhi,h^d=i,h^e=fgh,h^g=hj,i^a=fgij,i^b=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, #72258953
maximal normal subgroup of#72258941, #72258953

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, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.10);;
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, 1, F.6*F.10);;
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, 1, F.6*F.8*F.9);;
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, 1, F.6*F.7*F.9*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);