Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting, derived, Sylow | C3×H(5,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊D3×C3⋊(C3⋊D3) |
| commutator, Fitting, cosocle, hypocenter | C3×H(5,𝔽3) | C22 |
| socle | C32 | C3⋊D32 |
| LCS | G ▷ C3×H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
4336 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 4336 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 4 generators on 30 points:
(2,3)(5,16)(6,12)(8,9)(10,29)(11,18)(13,24)(14,23)(17,28)(19,20)(21,25)(22,27)(26,30);(2,3)(5,8)(6,13,14)(9,18)(10,26,22,17,19,25)(11,16)(12,23,24)(20,30,27,21,29,28);(1,2,3)(4,6,12)(5,10,21)(7,14,23)(8,17,20)(9,19,28)(11,22,30)(13,24,15)(16,25,29)(18,26,27);(4,5,8)(6,10,17)(7,11,16)(9,18,15)(12,21,20)(13,19,26)(14,22,25)(23,30,29)(24,28,27)Polycyclic presentation:
2^2,3^6|c^a=c^2,c^b=c^2,d^a=d^2,d^b=d^2,e^a=e^2,e^b=e^2,e^c=eh,f^b=f^2,g^b=g^2,g^f=gh102 conjugacy classes — view the full character table.
| maximal subgroup of | #72277838 |
| maximal normal subgroup of | #72277838 |
| maximal quotient of | #72277745 |
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> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);