Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C71 · Schur multiplier (H2) = C25 |
| direct product | C71×#11112 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #127 |
| commutator | C22×C4 | C22×C4×C71 |
| Frattini, cosocle | C23×C4 | C23×C71 |
| socle | C22×C71 | #399 |
| prime radical | C22×#41×C71 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C23×C4×C71 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 3 generators on 107 points:
(2,4)(5,6,10,14,11,15,9,25)(7,16,19,28,20,27,18,12)(8,21,23,13,24,30,22,31)(17,29,34,36,35,26,33,32)(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);(1,2,3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,28,25,27)(8,17,22,33,24,35,23,34)(13,29,30,36,31,26,21,32);(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,26)(14,30)(15,31)(16,32)(18,33)(19,34)(20,35)(21,25)(27,36)(28,29)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 4100);