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#72341865

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#392197 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72341801
commutator#85666C2×C4
FrattiniC23×C8C2×S4
Fitting#392197D3
prime radicalC2×#11715736C2
cosocleC2×#85666C22
hypercenterC2×C8#1020⋊C2
hypocenter#997#102
LCSG ▷ #85666 ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

1960 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1960 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,31,55,37,22,45,64,13,8,17,56,35,4,43,63,30)(2,29,49,24,9,20,61,60,14,6,50,48,11,38,62,52)(3,47,58,21,15,51,39,7,10,23,57,40,16,59,41,33)(5,34,53,46,42,26,28,18,25,36,54,44,19,12,32,27);(1,2,4,9,8,14,22,11)(3,5,15,19,10,25,16,42)(6,30,54,36,33,41,38,35,32,26,21,57,29,13,53,34,7,39,20,37,28,12,40,58)(17,48,62,59,27,56,45,52,50,47,44,63,31,24,61,51,18,55,43,60,49,23,46,64)

Polycyclic presentation:

2^5,3,2^5|a^2=c,b^2=cdek,c^2=k,d^2=e,e^2=k,g^2=k,h^2=k,b^a=bd,c^b=ck,d^a=de,d^b=dk,e^a=ek,f^b=f^2,g^a=gijk,g^b=hk,g^f=h,h^a=hik,h^b=gk,h^f=ghk,h^g=hk,i^b=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

70 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, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.4*F.5*F.11);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);