Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C71 · Schur multiplier (H2) = C22×C4 |
| direct product | C71×#9669 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #121 |
| commutator | C2×C4 | C22×C8×C71 |
| Frattini, cosocle | C2×C42 | C23×C71 |
| socle | C22×C71 | #390 |
| prime radical | C23×C71 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C2×C42×C71 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 9088 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,7,5,3,6,8)(9,10)(11,15,20,29)(12,22)(13,26)(14,17)(16,24)(18,23)(19,31,33,40)(21,35,32,28)(25,36)(27,38,39,34)(30,37)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,4,8,5,2,6,7)(9,11,14,20)(10,15)(12,19,25,33)(13,21,23,32)(16,27)(17,29)(18,28)(22,31)(24,34,37,38)(26,35)(30,39)(36,40);(9,12,23,37,14,25,13,24)(10,16,26,36,17,30,18,22)(11,19,32,38,20,33,21,34)(15,27,35,40,29,39,28,31)4544 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 2657);