Finite Group DatabasebetaFinite Group Databasebeta| order | 31242 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C41⋊(D127⋊C3) |
| unambiguous semidirect product | (C127⋊C3)⋊D41 |
| unambiguous semidirect product | D5207⋊C3 |
| Fitting, socle, derived, Hall | C41×C127 ⋊ C2×C3 |
| Hall | C41×C127⋊C3 ⋊ C2 |
| Hall | D5207 ⋊ C3 |
| Hall | C127⋊C3 ⋊ D41 |
| Sylow | C127 ⋊ C3×D41 |
| Sylow | C41 ⋊ D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C41×C127 | C2×C3 |
| LCS | G ▷ C41×C127 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 168 |
| transitive degree | μt(G) = 5207 |
A minimal faithful permutation representation — 2 generators on 168 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)(43,47,48,44,45,49)(46,55,56,50,51,65)(52,68,69,58,59,82)(53,67,63,61,57,54)(60,87,88,71,72,105)(62,85,86,74,70,79)(64,66,84,76,77,80)(73,110,111,90,91,133)(75,108,109,93,89,102)(78,83,107,95,96,103)(81,122,120,99,100,98)(92,138,139,113,114,159)(94,136,137,115,112,130)(97,106,135,117,118,131)(101,146,147,123,124,126)(104,152,144,127,128,121)(116,161,162,141,140,157)(119,134,160,143,142,158)(125,164,165,148,149,155)(129,168,167,153,154,151)(132,166,163,150,156,145);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(42,43,46,52,60,73,92,114,140,149,154,156,158,159,143,117,95,76,63,48,47,53,62,75,94,116,142,91,112,124,128,131,133,157,163,144,120,86,56,67,55,66,81,101,125,150,118,72,89,100,103,105,130,155,167,147,109,69,84,85,68,83,104,129,153,127,96,59,70,80,82,102,126,151,165,137,88,107,122,108,87,106,132,148,123,99,77,51,57,65,79,98,121,145,162,111,135,152,146,136,110,134,141,115,93,74,61,45,49,54,64,78,97,119,139,160,166,168,164,161,138,113,90,71,58,50,44)Polycyclic presentation:
2,3,41,127|c^a=c^40,d^a=d^126,d^b=d^107927 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31242, 12);