Finite Group DatabasebetaFinite Group Databasebeta| order | 8664 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #72227351 ⋊ C3 |
| Hall | #11762245 ⋊ D4 |
| Sylow | C192 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #14524206⋊C2 |
| commutator, Fitting, socle, cosocle | C2×C192 | C22×C3 |
| hypocenter | C192 | C3×D4 |
| LCS | G ▷ C2×C192 ▷ C192 | |
| UCS | C1 ◁ C2 | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 76 |
A minimal faithful permutation representation — 2 generators on 42 points:
(2,4)(6,10,11,7,8,12)(9,17,18,13,14,22)(15,23,21,20,19,16)(24,25,35,36,39,31)(26,32,28,30,41,40)(27,38,33,42,34,37);(1,2)(3,4)(5,6,9,14,19,22,21,11,10,15,20,8,12,16,18,23,17,13,7)(25,26)(27,29)(28,30)(31,32)(33,35)(34,36)(37,38)(39,41)(40,42)Polycyclic presentation:
2^2,3,2,19^2|b^a=bd,e^a=e^18,e^b=e^18,e^c=e^11,f^a=e^2f,f^b=f^18,f^c=f^1193 conjugacy classes — view the full character table.
| maximal subgroup of | #72227353, #72227355 |
| maximal normal subgroup of | #72227353, #72227355 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 19, 19]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 5, 2, F.5^18);;
gap> SetConjugate(coll, 5, 3, F.5^11);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^18);;
gap> SetConjugate(coll, 6, 3, F.6^11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);