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#72226048

order2304
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC2×#585 ⋊ C3×D3

Subgroups

subgroupG/H
centerC4C2×A4×S4
commutator, cosocle, hypocenter#994C23×C3
Frattini, socleC2C22×A4×S4
FittingC2×#585C3×D3
hypercenterC2×C4A4×S4
LCSG ▷ #994
UCSC1 ◁ C4 ◁ C2×C4

2246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 2246 conjugacy classes of subgroups — too many to draw. Only its 79 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:

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

Polycyclic presentation:

2^3,3,2^2,3,2^3|a^2=j,c^2=j,e^2=j,f^2=j,h^2=j,i^2=j,b^a=bj,c^a=cj,e^d=ef,f^d=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#72226249, #72226321, #72228062
maximal normal subgroup of#72226249, #72226321, #72228062

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, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, 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, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, 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);