Finite Group DatabasebetaFinite Group Databasebeta| order | 25074 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22552545 |
| Fitting, socle, derived | C3×C7×C199 ⋊ C2×C3 |
| Hall | C3×#16862007 ⋊ C2 |
| Hall | C7×C199 ⋊ C3×D3 |
| Sylow | C199 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ D3×C199⋊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:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,131)(98,121,141)(101,124,138)(104,127,135)(107,130,132)(110,133,129)(113,136,126)(116,139,123)(118,142,120)(122,146,154)(125,148,163)(128,151,160)(134,156,155)(137,158,153)(140,161,150)(143,164,147)(144,157,145)(149,168,173)(152,170,179)(159,175,174)(162,177,172)(165,180,169)(166,176,167)(171,184,186)(178,188,187)(181,190,185)(182,189,183)(191,195,194)(192,196,193)(197,199,198)(201,204,205,202,203,206)(208,209);(1,2,4,8,15,25,38,54,73,95,103,59,27,7,13,21,31,44,60,79,101,125,149,151,100,56,24,36,50,66,85,107,131,154,173,186,184,148,97,53,71,91,113,137,159,176,142,92,114,138,160,146,94,118,143,165,181,191,196,180,139,90,51,67,86,108,132,155,174,187,194,198,195,177,136,87,49,22,32,45,61,80,102,126,150,169,183,175,133,84,46,20,5,9,16,26,39,55,74,96,120,145,130,81,43,17,3,6,11,18,28,41,57,76,98,122,127,78,40,14,23,34,47,63,82,104,128,152,171,170,124,75,37,52,69,88,110,134,157,117,141,163,179,168,121,72,93,116,140,162,178,189,164,115,70,89,111,135,119,144,166,182,192,197,199,190,161,112,68,35,48,64,83,105,129,153,172,185,193,188,158,109,65,33,12,19,29,42,58,77,99,123,147,167,156,106,62,30,10)(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,d^b=d^2,e^b=e^106804 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25074, 60);