Finite Group DatabasebetaFinite Group Databasebeta| order | 32046 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | (C109⋊C3)⋊(C7⋊D7) |
| Fitting, socle, derived, Hall | C72×C109 ⋊ C2×C3 |
| Hall | C72×C109⋊C3 ⋊ C2 |
| Hall | C7⋊D763 ⋊ C3 |
| Hall | C109⋊C3 ⋊ C7⋊D7 |
| Sylow | C109 ⋊ C3×C7⋊D7 |
| Sylow | C72 ⋊ D109⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C109 | C2×C3 |
| LCS | G ▷ C72×C109 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5341 |
A minimal faithful permutation representation — 3 generators on 123 points:
(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)(16,20,21,17,18,22)(19,28,29,23,24,38)(25,41,42,31,32,55)(26,40,36,34,30,27)(33,60,61,44,45,78)(35,58,59,47,43,52)(37,39,57,49,50,53)(46,83,84,63,64,105)(48,81,82,66,62,75)(51,56,80,68,69,76)(54,94,92,72,73,71)(65,110,111,86,87,123)(67,108,109,88,85,102)(70,79,107,89,90,103)(74,115,116,95,96,98)(77,119,113,99,100,93)(91,106,117,112,97,122)(101,121,114,104,120,118);(8,9,11,13,14,12,10)(15,16,19,25,33,46,65,84,107,119,115,108,83,106,95,72,50,24,30,38,52,71,93,114,116,82,42,57,58,41,56,77,101,90,45,62,73,76,78,102,112,89,68,49,36,21,20,26,35,48,67,87,111,117,120,121,122,123,110,88,66,47,34,18,22,27,37,51,70,91,109,61,80,94,81,60,79,104,99,69,32,43,53,55,75,98,118,113,92,59,29,40,28,39,54,74,97,64,85,96,100,103,105,86,63,44,31,23,17);(1,2,4,6,7,5,3)Polycyclic presentation:
2,3,7^2,109|c^a=c^6,d^a=d^6,e^a=e^108,e^b=e^63960 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32046, 79);