Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C71 · Schur multiplier (H2) = C22 |
| direct product | C71×#7476 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C71 | #367 |
| commutator | C16 | C42×C71 |
| Frattini, cosocle | C22×C16 | C22×C71 |
| socle | C2×C71 | #1380 |
| prime radical | C23×C71 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C71 ◁ C22×C4×C71 ◁ C22×C8×C71 ◁ C22×C16×C71 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 2 generators on 135 points:
(1,2,5,12,9,15,8,34)(3,16,18,48,21,50,27,49)(4,13,23,43,26,46,25,60)(6,29,28,45,24,11,31,42)(7,32,22,44,30,41,33,14)(10,35,36,17,40,52,39,54)(19,57,55,38,53,47,61,63)(20,59,56,37,58,62,51,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3,6,19,30,58,23,35,5,27,28,61,33,56,26,54,9,21,24,53,7,20,25,52,8,18,31,55,22,51,4,17)(2,10,13,37,44,63,42,49,34,36,60,62,32,57,11,16,15,40,46,64,14,38,45,48,12,39,43,59,41,47,29,50)3692 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 464);