Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C71 · Schur multiplier (H2) = C2×C4 |
| direct product | C71×#13152 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | C2×D8 |
| commutator | C2×C8 | C22×C4×C71 |
| Frattini, cosocle | C22×C8 | C23×C71 |
| socle | C22×C71 | C4×D8 |
| prime radical | C2×#135×C71 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C22×C8×C71 ◁ C2×C4×C8×C71 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 3 generators on 107 points:
(1,2)(3,4)(5,6,11,17,12,19,30,18)(7,20,24,33,8,25,28,36)(9,29,31,15,21,16,10,32)(13,27,14,26,35,23,34,22)(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);(5,7,21,26,30,28,31,27,12,8,9,22,11,24,10,23)(6,13,15,20,17,14,16,33,19,35,32,25,18,34,29,36);(2,4)(5,8,9,26,30,24,10,27,12,7,21,22,11,28,31,23)(6,14,15,33,17,35,16,25,19,34,32,36,18,13,29,20)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 6140);