Finite Group DatabasebetaFinite Group Databasebeta| order | 35028 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26043019 |
| derived | C3×C7×C139 ⋊ C3×C4 |
| Hall | C3×#15668588 ⋊ C4 |
| Hall | C7×C139 ⋊ C3×Dic3 |
| Sylow | C139 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4) |
| Sylow | C7 ⋊ Dic3×C139⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19154250 |
| 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)(16,17,19)(18,21,26)(20,24,23)(22,28,36)(25,31,33)(27,34,30)(29,38,49)(32,41,46)(35,44,43)(37,47,40)(39,51,65)(42,54,62)(45,57,59)(48,60,56)(50,63,53)(52,67,84)(55,70,81)(58,73,78)(61,76,75)(64,79,72)(66,82,69)(68,86,105)(71,89,102)(74,92,100)(77,95,97)(80,98,94)(83,87,91)(85,103,88)(90,108,123)(93,111,121)(96,114,119)(99,117,116)(101,109,113)(104,122,110)(106,124,107)(112,127,137)(115,129,136)(118,132,134)(120,128,131)(125,138,126)(130,139,146)(133,141,145)(135,140,143)(142,147,151)(144,148,150)(149,152,153);(1,2,3)(8,9,11,14,13,12,10)(15,16,18,22,29,39,52,68,87,63,36,46,59,75,94,113,86,106,125,124,84,102,121,136,145,150,139,111,70,38,50,64,80,99,118,133,142,148,132,95,57,31,17,20,25,32,42,55,71,90,109,79,47,26,33,43,56,72,91,110,126,138,103,65,81,100,119,134,143,127,89,51,66,83,101,120,135,144,149,152,141,114,73,41,21,27,35,45,58,74,93,112,128,98,60,34,19,23,30,40,53,69,88,107,122,82,49,62,78,97,116,131,108,67,85,104,105,123,137,146,151,153,147,129,92,54,28,37,48,61,77,96,115,130,140,117,76,44,24)Polycyclic presentation:
2,3^2,7,139,2|a^2=f,c^a=c^2,d^a=d^6,d^b=d^2,e^b=e^421128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35028, 87);