Finite Group DatabasebetaFinite Group Databasebeta| order | 31098 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C71⋊(D73⋊C3) |
| unambiguous semidirect product | (C73⋊C3)⋊D71 |
| unambiguous semidirect product | D5183⋊C3 |
| Fitting, socle, derived, Hall | C71×C73 ⋊ C2×C3 |
| Hall | C71×C73⋊C3 ⋊ C2 |
| Hall | D5183 ⋊ C3 |
| Hall | C73⋊C3 ⋊ D71 |
| Sylow | C73 ⋊ C3×D71 |
| Sylow | C71 ⋊ D73⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C71×C73 | C2×C3 |
| LCS | G ▷ C71×C73 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 144 |
| transitive degree | μt(G) = 5183 |
A minimal faithful permutation representation — 2 generators on 144 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)(68,69)(70,71)(73,77,78,74,75,79)(76,85,86,80,81,95)(82,98,99,88,89,112)(83,97,93,91,87,84)(90,117,118,101,102,131)(92,115,116,104,100,109)(94,96,114,106,107,110)(103,119,128,105,120,134)(108,113,133,121,122,129)(111,138,136,125,126,124)(123,132,139,127,135,143)(130,144,142,140,141,137);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(72,73,76,82,90,103,104,91,75,79,84,94,108,123,125,107,81,87,95,109,124,137,140,122,89,100,110,112,128,139,144,135,102,119,126,129,131,118,133,138,120,117,132,141,143,134,99,114,115,98,113,130,142,136,116,86,97,85,96,111,127,121,106,93,78,77,83,92,105,101,88,80,74)Polycyclic presentation:
2,3,71,73|c^a=c^70,d^a=d^72,d^b=d^8963 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31098, 12);