Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C71 · Schur multiplier (H2) = C25 |
| direct product | C71×#23200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | D42 |
| commutator | C2×C4 | C23×C4×C71 |
| Frattini, cosocle | C22×C4 | C24×C71 |
| prime radical | C23×C71 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C22×C4×C71 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 4 generators on 107 points:
(1,2,4,3)(5,6,10,15,12,16,11,26)(7,17,19,30,20,29,25,13)(8,21,22,31,23,34,9,14)(18,27,24,32,33,36,35,28)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,3,4,2)(5,7,12,20)(6,13,16,30)(8,18,23,33)(9,24,22,35)(10,25,11,19)(14,27,31,36)(15,29,26,17)(21,28,34,32);(5,8,10,22,12,23,11,9)(6,14,26,34,16,31,15,21)(7,18,25,35,20,33,19,24)(13,27,17,32,30,36,29,28);(5,9,11,23,12,22,10,8)(6,14,26,34,16,31,15,21)(7,18,25,35,20,33,19,24)(13,28,29,36,30,32,17,27)4118 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 16188);