Finite Group DatabasebetaFinite Group Databasebeta| order | 29850 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C25⋊(D199⋊C3) |
| unambiguous semidirect product | (C199⋊C3)⋊D25 |
| unambiguous semidirect product | D4975⋊C3 |
| Fitting, derived, Hall | C25×C199 ⋊ C2×C3 |
| Hall | C25×C199⋊C3 ⋊ C2 |
| Hall | D4975 ⋊ C3 |
| Hall | C199⋊C3 ⋊ D25 |
| Sylow | C199 ⋊ C3×D25 |
| Sylow | C25 ⋊ D199⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C25×C199 | C2×C3 |
| Frattini | C5 | C5⋊(D199⋊C3) |
| socle | C5×C199 | C3×D5 |
| LCS | G ▷ C25×C199 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 224 |
| transitive degree | μt(G) = 4975 |
A minimal faithful permutation representation — 2 generators on 224 points:
(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(27,31,32,28,29,33)(30,39,40,34,35,49)(36,52,53,42,43,66)(37,51,47,45,41,38)(44,71,72,55,56,89)(46,69,70,58,54,63)(48,50,68,60,61,64)(57,94,95,74,75,118)(59,92,93,77,73,86)(62,67,91,79,80,87)(65,107,105,83,84,82)(76,123,124,97,98,153)(78,121,122,100,96,115)(81,90,120,102,103,116)(85,136,137,108,109,111)(88,142,134,112,113,106)(99,158,159,126,127,189)(101,156,157,129,125,150)(104,119,155,131,132,151)(110,168,169,138,139,146)(114,173,171,143,144,141)(117,178,166,147,148,135)(128,170,182,140,161,197)(130,192,193,162,160,186)(133,154,191,163,164,187)(145,200,201,174,175,177)(149,203,198,179,180,172)(152,208,195,183,184,167)(165,190,207,176,194,215)(181,220,218,204,205,202)(185,222,216,209,210,199)(188,221,206,212,213,196)(211,223,224,217,214,219);(1,15,25,18,4,7,21,19,10,13,16,14,12,11,20,8,6,5,24,17,3,2,22,23,9)(26,27,30,36,44,57,76,99,128,138,108,83,61,35,41,49,63,82,106,135,167,196,204,179,147,103,56,73,84,87,89,115,146,177,202,219,223,212,164,98,125,139,144,148,151,153,186,159,191,208,203,200,161,192,158,190,205,210,213,215,197,157,95,120,142,136,121,94,119,152,185,211,216,198,171,137,93,53,68,69,52,67,88,114,145,176,163,131,102,79,60,47,32,31,37,46,59,78,101,130,162,129,100,77,58,45,29,33,38,48,62,81,104,133,165,174,143,112,80,43,54,64,66,86,111,141,172,199,217,209,183,132,75,96,109,113,116,118,150,182,207,221,222,220,194,127,160,170,175,180,184,187,189,193,124,155,178,173,168,156,123,154,188,214,224,218,201,169,122,72,91,107,92,71,90,117,149,181,206,195,166,134,105,70,40,51,39,50,65,85,110,140,126,97,74,55,42,34,28)Polycyclic presentation:
2,3,5,199,5|c^5=e^4,c^a=c^4e,d^a=d^198,e^a=e^4,d^b=d^106867 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29850, 12);