Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C71 · Schur multiplier (H2) = C2×C4 |
| direct product | C71×#7144 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #95 |
| commutator | C2×C4 | C4×C8×C71 |
| Frattini, cosocle | C2×C4×C8 | C22×C71 |
| socle | C22×C71 | #336 |
| prime radical | C23×C71 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C2×C4×C8×C71 ◁ G | |
210 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 2 generators on 119 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,20,24,21,26,31,38,22,25,32,37,33,39,44,47)(19,27,29,40,30,41,43,48,28,23,34,35,42,36,45,46)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19)(18,23,25,27)(20,29)(21,30)(22,28)(24,35,37,40)(26,36,39,41)(31,43)(32,34)(33,42)(38,46,47,48)(44,45)4544 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 132);