Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C71 · Schur multiplier (H2) = C22 |
| direct product | C71×#7193 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #99 |
| commutator | C2×C4 | C4×C8×C71 |
| Frattini, cosocle | C22×M16 | C22×C71 |
| socle | C22×C71 | #324 |
| prime radical | C24×C71 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C22×C4×C71 ◁ C22×M16×C71 ◁ G | |
342 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2272 |
A minimal faithful permutation representation — 2 generators on 91 points:
(5,6,8,13,10,11,16,18,9,14,17,20,7,12,15,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,2,3,4)(6,11,14,12)(13,19,20,18)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 181);