Finite Group DatabasebetaFinite Group Databasebeta| order | 25074 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22552550 |
| Fitting, socle, derived | C3×C7×C199 ⋊ C2×C3 |
| Hall | C3×#16862006 ⋊ C2 |
| Hall | C7×C199 ⋊ C3×D3 |
| Sylow | C199 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3⋊(D199⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C199 | C2×C3 |
| LCS | G ▷ C3×C7×C199 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 209 |
| transitive degree | μt(G) = 4179 |
A minimal faithful permutation representation — 2 generators on 209 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,164)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,142,143,113,114,121)(89,149,147,118,119,116)(92,154,137,122,123,110)(103,159,160,136,127,186)(105,166,167,138,135,162)(108,129,165,140,141,163)(115,170,171,144,145,158)(120,175,176,150,151,153)(124,181,169,155,156,148)(139,187,188,168,161,190)(146,191,192,172,173,189)(152,193,194,177,178,185)(157,196,174,182,183,180)(179,197,198,195,184,199)(201,204,205,202,203,206)(208,209);(1,2,5,11,19,32,51,74,103,137,109,80,45,15,26,14,25,40,60,85,115,146,174,176,143,97,47,66,82,67,46,65,92,124,157,184,198,192,167,99,130,154,149,142,131,98,129,102,135,145,151,156,160,163,164,190,196,193,191,187,159,107,50,71,84,88,91,93,125,158,185,195,177,150,118,87,55,18,29,39,41,61,86,116,148,168,138,104,75,52,33,20,4,8,13,23,37,56,79,108,140,106,77,54,35,22,7,6,12,21,34,53,76,105,139,169,147,112,68,28,43,44,27,42,63,89,120,152,179,194,171,132,70,95,117,111,96,69,94,127,161,173,178,183,188,134,165,186,181,175,170,166,133,141,73,100,114,119,123,126,128,162,189,199,197,182,155,122,78,31,48,59,62,64,90,121,153,180,172,144,113,83,58,36,10,16,24,38,57,81,110,136,101,72,49,30,17,9,3)(200,201,203,206,205,204,202)(207,208,209)Polycyclic presentation:
2,3^2,7,199|c^a=c^2,d^a=d^6,e^a=e^198,d^b=d^2,e^b=e^92705 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25074, 57);