Finite Group DatabasebetaFinite Group Databasebeta| order | 35028 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26043023 |
| derived | C3×C7×C139 ⋊ C3×C4 |
| Hall | C3×#15668588 ⋊ C4 |
| Hall | C7×C139 ⋊ C3×Dic3 |
| Sylow | C139 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4) |
| Sylow | C7 ⋊ C3⋊((C139⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19154254 |
| commutator, hypocenter | C3×C7×C139 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7×C139 | C2×C3 |
| prime radical | C2×C3×#15668588 | C2 |
| LCS | G ▷ C3×C7×C139 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 153 |
| transitive degree | μt(G) = 11676 |
A minimal faithful permutation representation — 2 generators on 153 points:
(2,3)(4,5,6,7)(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)(46,83,84,63,64,107)(48,81,82,66,62,75)(51,56,80,68,69,76)(54,96,94,72,73,71)(65,112,113,86,87,137)(67,110,111,89,85,104)(70,79,109,91,92,105)(74,124,116,97,98,100)(77,128,122,101,102,95)(88,129,130,115,103,147)(90,139,140,117,114,135)(93,108,138,119,120,136)(99,143,142,125,126,132)(106,150,121,133,134,123)(118,148,149,141,131,152)(127,153,146,144,145,151);(1,2,3)(8,9,11,14,13,12,10)(15,16,19,25,33,46,65,88,116,82,42,57,58,41,56,77,103,131,145,149,113,138,150,147,143,139,112,133,92,45,62,73,76,78,104,132,141,117,89,66,47,34,18,22,27,37,51,70,93,121,122,94,59,29,40,28,39,54,74,99,127,146,140,84,109,128,124,110,83,108,120,64,85,98,102,105,107,135,151,144,125,97,72,50,24,30,38,52,71,95,123,119,91,68,49,36,21,20,26,35,48,67,90,118,142,111,61,80,96,81,60,79,106,87,114,126,130,134,136,137,152,153,148,129,101,69,32,43,53,55,75,100,115,86,63,44,31,23,17)Polycyclic presentation:
2,3^2,7,139,2|a^2=f,c^a=c^2,d^a=d^6,e^a=e^138,d^b=d^2,e^b=e^42990 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35028, 85);