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#72228046

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72209660
commutator, cosocle, hypocenter#72227697C22
FrattiniC22×Q8C2×#1020
Fitting#72227698D3
prime radical#72227699C2
hypercenterC23#87588⋊C2
LCSG ▷ #72227697
UCSC1 ◁ C22 ◁ C23

4496 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 4496 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 144
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 2 generators on 144 points:

(1,6,4,7)(2,15,9,5)(3,11,12,10)(8,14,16,13)(17,39,54,110,25,114,51,99)(18,129,28,65,32,89,83,90)(19,53,100,23,40,64,34,57)(20,35,124,102,48,106,120,96)(21,98,22,38,55,104,59,62)(24,56,52,50,58,108,61,36)(26,82,95,86,33,49,78,31)(27,140,79,93,77,133,72,68)(29,69,85,132,87,88,80,130)(30,70,81,92,84,94,128,67)(37,46,105,121,107,63,97,123)(41,137,143,139,73,135,116,111)(42,122,118,45,113,126,144,60)(43,109,44,119,112,103,47,125)(66,115,141,74,91,142,138,75)(71,127,131,76,101,136,134,117);(2,6)(3,12,4)(7,10,8,9,16,11)(13,14,15)(17,18)(19,33)(20,115,55,41,43,82)(21,73,112,49,48,142)(22,87,24,31,57,30)(23,84,59,29,58,86)(25,32)(26,40)(27,122,83,45,72,51)(28,60,79,54,77,126)(34,131,106,95,96,71)(35,78,102,101,100,134)(36,139,42,90,107,66)(37,91,50,111,113,65)(38,94,108,88,39,89)(44,136,53,74,123,80)(46,116,52,117,120,81)(47,127,64,75,121,85)(56,69,114,129,62,70)(61,76,124,128,63,143)(67,109,137,110,93,105)(68,97,92,103,135,99)(98,138,119,132,144,133)(104,141,125,130,118,140)

Polycyclic presentation:

2^2,3,2^9|b^2=j,d^2=j,e^2=j,f^2=l,g^2=l,h^2=l,i^2=l,b^a=bjl,c^b=c^2,d^a=dhjk,d^b=ej,d^c=de,e^a=eijk,e^b=dj,e^c=d,e^d=ej,f^a=fhijk,f^b=efg,f^c=g,f^d=fj,g^a=gijk,g^b=degl,g^c=fgl,g^e=gj,g^f=gl,h^b=il,h^c=hi,h^d=hl,h^e=hjk,h^f=hjkl,h^g=hjk,i^b=hl,i^c=h,i^d=ijkl,i^e=il,i^f=ijk,i^g=il,i^h=il,k^a=kl,k^b=kl

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228055

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.10*F.12);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.5*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.4);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.4*F.5*F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.9*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.8*F.12);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);