Finite Group DatabasebetaFinite Group Databasebeta| order | 35028 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26043018 |
| derived | C3×C7×C139 ⋊ C3×C4 |
| Hall | C3×#15668589 ⋊ 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⋊#19154249 |
| commutator, hypocenter | C3×C7×C139 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7×C139 | C2×C3 |
| prime radical | C2×C3×#15668589 | 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)(74,87,99)(77,94,96)(80,97,93)(83,100,91)(85,103,88)(90,108,123)(92,109,120)(95,113,117)(98,115,114)(101,118,112)(104,121,110)(106,124,107)(111,127,136)(116,130,131)(119,132,129)(122,134,128)(125,137,126)(133,142,141)(135,143,140)(138,145,139)(144,149,148)(146,150,147)(151,153,152);(1,2,3)(8,9,11,14,13,12,10)(15,16,18,22,29,39,52,68,87,54,28,37,48,61,77,95,105,123,136,108,67,85,104,122,135,144,145,121,82,49,62,78,96,114,129,140,147,149,132,97,60,34,19,23,30,40,53,69,88,107,113,73,41,21,27,35,45,58,74,92,111,127,89,51,66,83,101,119,133,137,103,65,81,99,117,131,141,148,152,153,143,118,79,47,26,33,43,56,72,91,110,126,130,94,57,31,17,20,25,32,42,55,71,90,109,70,38,50,64,80,98,116,124,84,102,120,86,106,125,138,146,151,150,134,100,63,36,46,59,75,93,112,128,139,142,115,76,44,24)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^961128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35028, 88);