Finite Group DatabasebetaFinite Group Databasebeta| order | 37278 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27159100 |
| Fitting, socle, derived | C3×C19×C109 ⋊ C2×C3 |
| Hall | C3×#20261879 ⋊ C2 |
| Hall | C19×C109 ⋊ C3×D3 |
| Sylow | C109 ⋊ C3⋊(D19⋊C3) |
| Sylow | C19 ⋊ C3⋊(D109⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C19×C109 | C2×C3 |
| LCS | G ▷ C3×C19×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 6213 |
A minimal faithful permutation representation — 2 generators on 131 points:
(2,3)(5,9,10,6,7,11)(8,17,14,12,13,19)(15,20,22,21,18,16)(24,28,29,25,26,30)(27,36,37,31,32,46)(33,49,50,39,40,63)(34,48,44,42,38,35)(41,68,69,52,53,86)(43,66,67,55,51,60)(45,47,65,57,58,61)(54,91,92,71,72,114)(56,89,90,74,70,83)(59,64,88,76,77,84)(62,103,101,80,81,79)(73,118,100,94,95,120)(75,116,117,96,93,111)(78,87,115,98,99,112)(82,123,124,104,105,107)(85,127,121,108,109,102)(97,130,122,113,119,131)(106,128,126,110,125,129);(1,2,3)(4,5,8,14,20,17,21,7,11,16,22,10,9,15,13,18,19,12,6)(23,24,27,33,41,54,73,72,93,105,109,112,114,131,121,101,67,37,48,36,47,62,82,106,108,77,40,51,61,63,83,107,126,117,69,88,103,89,68,87,113,96,74,55,42,26,30,35,45,59,78,100,95,119,125,128,130,118,120,98,76,57,44,29,28,34,43,56,75,97,99,53,70,81,84,86,111,129,124,90,50,65,66,49,64,85,110,104,80,58,32,38,46,60,79,102,122,92,115,127,123,116,91,94,71,52,39,31,25)Polycyclic presentation:
2,3^2,19,109|c^a=c^2,d^a=d^18,e^a=e^108,d^b=d^7,e^b=e^451044 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37278, 94);