Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8×C67 · Schur multiplier (H2) = C29 |
| direct product | C2×C8×#41×C67 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8×C67 | C22 |
| commutator | C2 | C24×C8×C67 |
| Frattini, cosocle | C2×C4 | C25×C67 |
| socle | C23×C67 | C23×C4 |
| prime radical | C22×#41×C67 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8×C67 ◁ G | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 8576 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(11,12,13,15,14,16,17,18);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10);(3,5,6,9)(4,8,7,10)10720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 53184);