Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | #10624749 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #13091120 |
| commutator, hypocenter | #10624749 | C4 |
| Frattini | C2×C33 | C3⋊(C3⋊D3) |
| Fitting, prime radical, cosocle | C2×#10624749 | C2 |
| socle | C2×C3 | #92338 |
| LCS | G ▷ #10624749 | |
| UCS | C1 ◁ C2 | |
548 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4)(6,10)(7,13)(8,9)(11,20)(12,23)(14,25)(15,17)(16,19)(18,24)(21,27)(22,30)(26,28)(29,31);(5,14,13,9,17,15,8,7,25)(6,22,24,12,26,21,16,11,31)(10,29,20,19,27,28,23,18,30);(7,14,17)(13,15,25)(18,27,29)(21,24,31);(5,6,10)(7,11,18)(8,16,23)(9,12,19)(13,24,20)(14,22,29)(15,21,28)(17,26,27)(25,31,30)Polycyclic presentation:
2^2,3^6|a^2=b,e^3=h,c^a=c^2,d^a=d^2,d^c=dg,e^a=e^2h^2,e^d=ef,f^a=fh^2,f^c=fh,f^e=fh,g^e=gh^2,h^a=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#13091120, #72228545, #72229214, … |
| maximal normal subgroup of | C4×#13091120, #72228545, #72229214, … |
| maximal quotient of | #72247336 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);