Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C71 · Schur multiplier (H2) = C2 |
| direct product | C71×#13108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C71 | #387 |
| commutator | C22×C4 | C22×C4×C71 |
| Frattini, cosocle | C22×C8 | C23×C71 |
| socle | C2×C71 | #1917 |
| prime radical | C23×C71 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C71 ◁ C22×C8×C71 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 2272 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2,6,13,7,11,25,30,8,12,26,14,5,21,23,17)(3,4,10,20,15,9,27,28,16,19,29,24,22,18,32,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3,6,10,7,15,25,27,8,16,26,29,5,22,23,32)(2,9,13,28,11,19,30,24,12,18,14,31,21,4,17,20);(1,4,5,18,8,19,7,9)(2,10,11,27,12,29,21,32)(3,14,15,17,16,13,22,30)(6,24,23,28,26,20,25,31)3266 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 6096);