Finite Group DatabasebetaFinite Group Databasebeta| order | 37278 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#27159096 |
| Fitting, socle, derived | C3×C19×C109 ⋊ C2×C3 |
| Hall | C3×#20261879 ⋊ C2 |
| Hall | C19×C109 ⋊ C3×D3 |
| Sylow | C109 ⋊ C3⋊(D19⋊C3) |
| Sylow | C19 ⋊ D3×C109⋊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)(24,25,27)(26,29,34)(28,32,31)(30,36,44)(33,39,41)(35,42,38)(37,46,57)(40,49,54)(43,52,51)(45,55,48)(47,59,73)(50,62,70)(53,65,67)(56,68,64)(58,71,61)(60,75,78)(63,77,88)(66,80,85)(69,83,82)(72,86,79)(74,89,76)(81,94,102)(84,97,99)(87,100,96)(90,103,93)(91,95,92)(98,108,113)(101,111,110)(104,114,107)(105,109,106)(112,119,121)(115,122,118)(116,120,117)(123,127,126)(124,128,125)(129,131,130);(1,2,3)(4,5,8,14,20,17,21,7,11,16,22,10,9,15,13,18,19,12,6)(23,24,26,30,37,47,60,59,74,90,104,115,123,128,114,86,55,34,41,51,64,79,93,106,94,62,36,45,56,69,84,98,109,89,57,70,85,99,110,118,125,119,97,65,39,25,28,33,40,50,63,78,75,91,105,116,124,129,131,122,100,68,42,27,31,38,48,61,76,92,77,46,58,72,87,101,112,120,103,71,44,54,67,82,96,107,117,108,80,49,29,35,43,53,66,81,95,73,88,102,113,121,126,130,127,111,83,52,32)Polycyclic presentation:
2,3^2,19,109|c^a=c^2,d^a=d^18,d^b=d^7,e^b=e^451098 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37278, 96);