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#72226069

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC22C2×A4⋊S4
commutator, hypocenterΩ+(4,𝔽3)C2×C4
FittingC22×H(5,𝔽2)C3⋊D3
prime radicalC22×Ω+(4,𝔽3)C2
cosocleC2×Ω+(4,𝔽3)C22
hypercenterC23A4⋊S4
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C23

948 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 52
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 52 points:

(5,22,52,27,21,40,10,7,42,24,18,47)(6,33,38,23,51,37,11,50,46,14,32,44)(8,19,17,29,20,41,25,45,43,9,28,48)(12,16,31,49,39,26,30,35,15,34,36,13);(1,2,3,4)(5,6)(7,16)(8,23)(9,26)(10,11)(12,17)(13,29)(14,25)(15,24)(18,44)(19,39)(20,46)(21,37)(22,35)(27,31)(28,38)(30,43)(32,48)(33,47)(34,52)(36,45)(40,50)(41,51)(42,49);(8,9,27)(13,14,15)(18,19,28)(20,21,45)(23,31,26)(24,25,29)(32,33,34)(36,37,46)(38,39,44)(40,41,42)(47,48,52)(49,51,50)

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,f^2=j,g^2=j,b^a=bj,d^a=d^2,e^a=e^2,f^a=gj,f^d=g,f^e=g,g^a=fj,g^d=fgj,g^e=fgj,g^f=gj,h^a=fhi,h^d=i,h^e=fgh,h^g=hj,i^a=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#72226133, #72228357, #72228367
maximal normal subgroup of#72226133, #72228357, #72228367

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, 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, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, 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, 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, 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, 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);