Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C67 · Schur multiplier (H2) = C214 |
| direct product | C2×C4×H(5,𝔽2)×C67 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C67 | C24 |
| commutator | C2 | C25×C4×C67 |
| Frattini, cosocle | C22 | C26×C67 |
| socle | C23×C67 | C25 |
| prime radical | C22×H(5,𝔽2)×C67 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C67 ◁ G | |
12378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 4288 |
A minimal faithful permutation representation — 6 generators on 81 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(3,4,5,6);(7,8)(9,13)(10,12)(11,14);(7,9)(8,11)(10,14)(12,13);(9,14)(11,13);(8,12)(11,13)9112 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 55637);