Finite Group DatabasebetaFinite Group Databasebeta| order | 27180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| Fitting, derived, Hall | C9×C151 ⋊ C22×C5 |
| Hall | C9×C151⋊C5 ⋊ C22 |
| Hall | D9×D151 ⋊ C5 |
| Hall | C151⋊C5 ⋊ C2×D9 |
| Sylow | C151 ⋊ C2×C5×D9 |
| Sylow | C9 ⋊ C2×D151⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C9×C151 | C22×C5 |
| Frattini | C3 | D3×D151⋊C5 |
| socle | C3×C151 | C2×C5×D3 |
| LCS | G ▷ C9×C151 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 160 |
| transitive degree | μt(G) = 1359 |
A minimal faithful permutation representation — 2 generators on 160 points:
(2,6,7,25,30,3,4,12,14,53)(5,19,20,84,88,9,10,41,43,124)(8,35,36,37,64,16,17,74,75,77)(11,48,49,29,102,22,23,81,61,62)(13,58,59,116,83,27,28,73,105,100)(15,68,69,65,66,32,33,70,31,76)(18,79,80,142,129,38,39,121,122,111)(21,93,94,24,97,45,46,133,50,51)(26,101,63,54,52,55,56,108,92,98)(34,114,113,130,123,71,72,144,89,149)(40,86,87,136,137,82,42,127,96,151)(44,131,85,110,95,90,91,126,67,47)(57,60,139,125,78,99,103,106,150,119)(104,145,107,128,138,140,120,132,141,118)(109,112,146,147,148,143,134,115,117,135)(152,157,155,154,159,156,158,153,160);(1,2,5,11,24,52,99,55,7,15,34,73,117,146,114,126,20,44,92,14,31,67,111,124,49,79,23,36,76,118,136,95,45,19,42,80,123,150,43,89,56,102,137,87,128,116,94,135,106,142,140,28,60,107,131,143,108,61,27,4,8,18,40,83,30,64,97,17,37,78,120,144,69,68,112,46,93,134,33,70,113,145,119,75,35,51,77,53,100,82,38,16,6,13,29,63,109,91,132,103,58,104,122,139,148,133,105,141,127,151,62,101,130,84,125,149,121,86,10,21,47,96,138,66,74,48,39,81,88,129,110,65,25,54,90,41,85,72,115,147,59,71,32,12,26,57,98,50,22,9,3)(153,155)(154,158)(156,159)(157,160)150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27180, 21);