Finite Group DatabasebetaFinite Group Databasebeta| order | 24522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C61⋊C3)⋊D67 |
| Fitting, socle, derived, Hall | C61×C67 ⋊ C2×C3 |
| Hall | C67×C61⋊C3 ⋊ C2 |
| Hall | D4087 ⋊ C3 |
| Hall | C61⋊C3 ⋊ D67 |
| Sylow | C67 ⋊ D61⋊C3 |
| Sylow | C61 ⋊ C3×D67 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C61×C67 | C2×C3 |
| LCS | G ▷ C61×C67 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 4087 |
A minimal faithful permutation representation — 2 generators on 128 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(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)(69,73,74,70,71,75)(72,81,82,76,77,91)(78,94,95,84,85,107)(79,93,89,87,83,80)(86,112,113,97,98,123)(88,110,111,99,96,104)(90,92,109,101,102,105)(100,125,126,115,114,121)(103,108,124,117,116,122)(106,128,127,119,120,118);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(68,69,72,78,86,98,114,120,122,123,117,101,89,74,73,79,88,100,116,85,96,105,107,121,127,111,82,93,81,92,106,119,102,77,83,91,104,118,126,95,109,110,94,108,115,99,87,71,75,80,90,103,113,124,128,125,112,97,84,76,70)Polycyclic presentation:
2,3,61,67|c^a=c^60,d^a=d^66,c^b=c^47775 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24522, 22);