Finite Group DatabasebetaFinite Group Databasebeta| order | 17152 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C67 · Schur multiplier (H2) = C22 |
| direct product | C67×#7397 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C67 | #353 |
| commutator | C2×M16 | C2×C4×C67 |
| Frattini, cosocle | #407 | C22×C67 |
| socle | C2×C67 | #1343 |
| prime radical | C67×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C67 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ #407×C67 ◁ G | |
268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 2144 |
A minimal faithful permutation representation — 2 generators on 99 points:
(1,2,5,10,7,12,6,11)(3,13,15,29,17,30,16,26)(4,18,20,27,21,28,22,9)(8,19,23,14,25,31,24,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,24)(11,23)(12,25)(13,18)(14,20)(21,31)(22,32)(27,29)(28,30)2077 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17152, 385);