Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C71 · Schur multiplier (H2) = C25 |
| direct product | C71×#17716 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C71 |
| Frattini, cosocle | C42 | C24×C71 |
| socle | C22×C71 | C22×C22⋊C4 |
| prime radical | C71×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C22×C42×C71 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,5,8,6,9,4,13)(3,10,11,15,12,7,14,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,32,31);(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)4544 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 10704);