Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C71 · Schur multiplier (H2) = C23 |
| direct product | C71×#11597 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C71 | #102 |
| commutator | C8 | C22×C8×C71 |
| Frattini, cosocle | C4×C8 | C23×C71 |
| socle | C22×C71 | C2×#102 |
| prime radical | C2×#135×C71 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C71 ◁ C2×C42×C71 ◁ C2×C4×C8×C71 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 9088 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,37)(12,16)(13,27)(14,19)(15,22,31,39)(17,25)(18,26)(21,36,24,33)(28,35,40,30)(29,32)(34,38)(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);(9,11)(10,15)(12,21)(13,28)(14,23)(16,30)(17,33)(18,35)(19,31)(20,32)(22,38)(24,26)(25,39)(27,37)(29,40)(34,36);(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,38)(15,30,31,35)(17,32,34,27)(20,36,37,33)(22,28,39,40)4544 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 4585);