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#72236914

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#11713201
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#60399D3
prime radical#60399⋊C3C2
hypercenterD4#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ D4

1210 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1210 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,17,26,13,7,19,15,23)(2,28,31,25,11,29,12,6)(3,24,9,27,14,20,30,16)(4,18,5,22,8,32,21,10);(1,3,7,14)(2,8,11,4)(5,18,21,32)(6,23,25,13)(9,29,30,28)(10,16,22,27)(12,24,31,20)(15,17,26,19);(1,2,4,3,7,11,8,14)(5,17,15,18,21,19,26,32)(6,22,16,23,25,10,27,13)(9,28,12,24,30,29,31,20)

Polycyclic presentation:

2^4,3,2^5|a^2=d,b^2=j,c^2=d,d^2=j,f^2=j,g^2=j,b^a=bdj,c^b=cd,d^b=dj,e^a=e^2,e^b=e^2,e^c=e^2,f^a=fg,f^b=fg,f^c=fgij,f^e=g,g^a=gj,g^b=gj,g^c=gh,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^a=hi,i^b=hi,i^c=hi,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240979, #72240992, #72240995, …
maximal normal subgroup of#72240979, #72240992, #72240995, …
maximal quotient ofC2×#72236914, #72343900

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