Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C67 · Schur multiplier (H2) = C27 |
| direct product | C67×#20989 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C67 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C67 |
| Frattini, cosocle | C22×C4 | C24×C67 |
| socle | C22×C67 | C22×C22⋊C4 |
| prime radical | C2×#525×C67 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C67 ◁ C24×C4×C67 ◁ G | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 2144 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2)(3,4)(5,6,9,14,10,13,8,12)(7,11,15,18,16,19,17,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(5,7,9,17,10,16,8,15)(6,11,12,18,13,19,14,20);(2,4);(5,8,10,9)(6,12,13,14)(7,15,16,17)(11,18,19,20)4288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 13977);