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#72341149

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Splittings

FittingC4×#3549 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#11715754
commutator, cosocleC2×#5464C23
Frattini, socleC23C22×#1020
FittingC4×#3549D3
hypercenterC4#11715754
hypocenter#5464C2×D4
LCSG ▷ C2×#5464 ▷ #5464
UCSC1 ◁ C2 ◁ C4

5534 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 5534 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^4,3,2^6|c^2=d,h^2=k,i^2=jk,c^a=cd,e^a=e^2,f^a=fgjk,f^b=fj,f^e=g,g^a=gk,g^b=gk,g^e=fg,h^a=fij,h^b=hk,h^e=ik,h^f=hk,h^g=hj,i^a=fgh,i^b=ijk,i^e=hi,i^f=ij,i^g=ijk,j^a=jk,j^e=k,k^e=jk

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341172, C2×#72341149
maximal normal subgroup of#72341172, C2×#72341149
maximal quotient ofC2×#72341149

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