Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×#2656 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#857⋊C2 |
| commutator, cosocle, hypocenter | #95⋊C3 | C24 |
| Frattini, socle | C23 | C23×S4 |
| Fitting | C2×#2656 | D3 |
| hypercenter | C23 | #857⋊C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C23 | |
3060 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 3060 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,6,11)(2,12,4)(3,7,10)(5,8,9)(13,17,16,18,31,28)(14,35,21)(15,25,24,23,38,37)(19,43,33)(20,22,26)(27,30)(32,34,39)(40,42);(2,6)(5,10)(7,9)(11,12)(15,23)(19,32)(24,37)(25,38)(33,39)(34,43)(40,42)(41,44);(3,8)(7,9)(11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,32)(21,33)(22,34)(26,39)(27,40)(28,37)(29,41)(30,42)(31,38)(35,43)(36,44);(2,5)(6,10)(7,12)(9,11)(13,14)(15,19)(16,26)(17,29)(18,20)(21,28)(22,27)(23,32)(24,39)(25,41)(30,35)(31,36)(33,37)(34,40)(38,44)(42,43)Polycyclic presentation:
2^4,3,2^5|f^2=h,g^2=i,c^b=cj,d^a=dj,e^a=e^2,e^c=ehi,f^a=fj,f^b=fhj,f^e=ghj,g^a=fghi,g^b=gij,g^e=fgj,g^f=gj,h^e=i,i^a=hij,i^e=hij60 conjugacy classes — view the full character table.
| maximal subgroup of | #72258516 |
| maximal normal subgroup of | #72258516 |
| maximal quotient of | #72228208 |
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, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
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, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);