Finite Group DatabasebetaFinite Group Databasebeta| order | 32046 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C109 ⋊ C2×C3 |
| Hall | #30630923 ⋊ C2 |
| Hall | C7⋊D763 ⋊ C3 |
| Sylow | C109 ⋊ #64424 |
| 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:
(2,5,6,3,4,7)(9,12,13,10,11,14)(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)(46,83,84,63,64,106)(48,81,82,66,62,75)(51,56,80,68,69,76)(54,95,93,72,73,71)(65,110,92,86,87,112)(67,108,109,88,85,103)(70,79,107,90,91,104)(74,115,116,96,97,99)(77,119,113,100,101,94)(89,122,114,105,111,123)(98,120,118,102,117,121);(8,9,11,14,13,12,10)(15,16,19,25,33,46,65,64,85,97,101,104,106,123,113,93,59,29,40,28,39,54,74,98,100,69,32,43,53,55,75,99,118,109,61,80,95,81,60,79,105,88,66,47,34,18,22,27,37,51,70,92,87,111,117,120,122,110,112,90,68,49,36,21,20,26,35,48,67,89,91,45,62,73,76,78,103,121,116,82,42,57,58,41,56,77,102,96,72,50,24,30,38,52,71,94,114,84,107,119,115,108,83,86,63,44,31,23,17);(1,2,4,7,6,5,3)Polycyclic presentation:
2,3,7^2,109|c^a=c^6,d^a=d^6,e^a=e^108,c^b=c^2,d^b=d^2,e^b=e^45896 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32046, 57);