Finite Group DatabasebetaFinite Group Databasebeta| order | 32046 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C49⋊C3)⋊D109 |
| Fitting, derived, Hall | C49×C109 ⋊ C2×C3 |
| Hall | C109×C49⋊C3 ⋊ C2 |
| Hall | D5341 ⋊ C3 |
| Hall | C49⋊C3 ⋊ D109 |
| Sylow | C109 ⋊ D49⋊C3 |
| Sylow | C49 ⋊ C3×D109 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C109 | C2×C3 |
| Frattini | C7 | ΓL(1,𝔽8)⋊D109 |
| socle | C7×C109 | AGL(1,𝔽7) |
| LCS | G ▷ C49×C109 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 158 |
| transitive degree | μt(G) = 5341 |
A minimal faithful permutation representation — 2 generators on 158 points:
(2,10,12,4,5,13)(3,22,24,8,9,25)(6,34,21,15,16,43)(7,39,40,19,20,49)(11,18,17,23,28,36)(14,37,46,32,33,47)(26,38,41,44,27,30)(29,42,31,45,35,48)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158);(1,23,20,31,30,35,4,8,45,14,13,40,15,19,22,46,5,34,36,28,43,24,44,38,37,33,27,26,25,21,18,17,16,10,47,9,7,6,49,12,32,29,3,2,42,41,48,39,11)(50,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,157,158,156,154,152,150,148,146,144,142,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52)Polycyclic presentation:
2,3,7,109,7|c^7=e^6,c^a=c^6e,d^a=d^108,e^a=e^6,c^b=c^2e^3,e^b=e^21040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32046, 22);