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#72226130

order4608
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

derivedΩ+(4,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, Frattini, socleC22C2×S42
commutator, hypocenterΩ+(4,𝔽3)C22×C4
FittingC22×H(5,𝔽2)D32
prime radicalC22×SO+(4,𝔽3)C2
cosocleC2×Ω+(4,𝔽3)C23
hypercenterC23S42
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C23

3148 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 3148 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,g^2=k,h^2=k,c^a=ck,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=gk,g^b=gk,g^e=gh,g^f=h,h^a=ghk,h^b=ghk,h^e=g,h^f=gh,h^g=hk,i^a=gi,i^b=ghj,i^e=gik,i^f=j,i^g=ik,i^h=ik,j^a=jk,j^b=hi,j^e=hjk,j^f=ij,j^h=jk

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);