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#72226013

order2304
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, FrattiniC4C2×A4×S4
commutator, hypocenter#994C2×C3×C4
FittingC2×#585C3×D3
socleC2C2×A4×SL(2,ℤ/4)
prime radicalC2×(#41⋊C3)⋊A4C2
cosocle#5945C22×C3
hypercenterC2×C4A4×S4
LCSG ▷ #994
UCSC1 ◁ C4 ◁ C2×C4

838 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 2 generators on 96 points:

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

Polycyclic presentation:

2^2,3,2^3,3,2^3|a^2=d,d^2=j,e^2=j,f^2=j,h^2=j,i^2=j,b^a=bj,e^c=ef,f^c=e,f^e=fj,g^a=g^2,h^a=ij,h^g=i,i^a=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226197, #72226249, #72226321
maximal normal subgroup of#72226197, #72226249, #72226321

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