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#72226016

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, hypercenterC4S42
commutator, hypocenterΩ+(4,𝔽3)C2×C4
Fitting#585D32
socleC2#11923458
prime radical#11923520C2
cosocle(#41⋊C3)⋊A4C22
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C4

550 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 29 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:

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,c^2=j,f^2=j,g^2=j,b^a=bj,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=fj,f^b=fj,f^d=fg,f^e=g,g^a=fgj,g^b=fgj,g^d=f,g^e=fg,g^f=gj,h^a=fh,h^b=fgi,h^d=fhj,h^e=i,h^f=hj,h^g=hj,i^a=ij,i^b=gh,i^d=gij,i^e=hi,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226197, #72226297
maximal normal subgroup of#72226197, #72226297

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