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#72228093

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

Aliases

No aliases for this group.

Splittings

Fitting#3573 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, FrattiniC4C2×A4⋊S4
commutator, hypocenterΩ+(4,𝔽3)C2×C4
Fitting#3573C3⋊D3
socleC2C4×A4⋊S4
prime radical#11923520C2
cosocle(#41⋊C3)⋊A4C22
hypercenterC8A4⋊S4
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C4 ◁ C8

633 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 39 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)(3,22)(4,60,31,64)(5,28,8,6)(7,29)(12,80,47,81)(13,14,16,34)(17,85,56,89)(18,53,21,19)(20,54)(23,26,57,24)(25,63,27,58)(30,59,61,62)(33,78,44,82)(36,94,75,95)(37,38,40,66)(41,45,77,43)(42,68,79,46)(48,51,83,49)(50,88,52,84)(55,76,86,87)(65,92,72,96)(69,73,91,71)(70,90,93,74);(1,12,48,11,27,41,22,65,23,2,17,70,9,33,76,32,4,42,3,36,59,10,52,69)(5,44,88,37,31,82,21,75,63,13,50,95,8,47,89,40,25,81,18,72,64,16,56,96)(6,45,83,38,61,77,53,90,57,14,51,93,28,68,87,66,26,79,19,73,62,34,86,91)(7,46,85,39,24,80,54,71,60,15,55,92,29,43,84,67,30,78,20,74,58,35,49,94);(5,6,29)(7,8,28)(13,14,35)(15,16,34)(18,19,54)(20,21,53)(24,25,26)(30,31,61)(37,38,67)(39,40,66)(43,44,45)(46,47,68)(49,50,51)(55,56,86)(57,58,63)(60,64,62)(71,72,73)(74,75,90)(77,78,82)(79,80,81)(83,84,88)(85,89,87)(91,92,96)(93,94,95)

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,c^2=j,f^2=j,g^2=j,b^a=bj,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

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228306, #72258953
maximal normal subgroup of#72228306, #72258953
maximal quotient of#72253043

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.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);