Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C71 · Schur multiplier (H2) = C22×C4 |
| direct product | C71×#13570 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C71 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C71 |
| Frattini, cosocle | #124 | C23×C71 |
| prime radical | C71×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C22×C4×C71 ◁ #532×C71 ◁ G | |
482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 2272 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2)(3,13,16,23)(4,9)(5,20)(6,11,22,28)(7,12)(8,15,24,32)(10,21,30,19)(14,25,31,26)(17,27,18,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3)(2,8)(4,15)(5,14)(6,16)(7,18)(9,27)(10,29)(11,24)(12,26)(13,30)(17,22)(19,32)(20,23)(21,31)(25,28);(1,4,7,5)(2,9,12,20)(3,14,18,15)(6,19,22,21)(8,23,26,27)(10,28,30,11)(13,25,29,24)(16,31,17,32)1988 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 6558);