Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C71 · Schur multiplier (H2) = C24 |
| direct product | C71×#11230 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #117 |
| commutator | C22×C4 | C22×C4×C71 |
| Frattini, cosocle | C23×C4 | C23×C71 |
| socle | C22×C71 | #380 |
| prime radical | C24×C71 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C22×M16×C71 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 3 generators on 107 points:
(5,6,10,14,11,16,9,15)(7,17,20,28,21,27,19,12)(8,22,23,30,24,29,25,13)(18,26,32,35,33,36,34,31)(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)(5,7,9,19,11,21,10,20)(6,12,14,17,16,28,15,27)(8,18,23,32,24,33,25,34)(13,26,29,31,30,36,22,35);(2,4)(5,8,9,23,11,24,10,25)(6,13,14,29,16,30,15,22)(7,18,19,32,21,33,20,34)(12,26,17,31,28,36,27,35)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 4218);