Finite Group DatabasebetaFinite Group Databasebeta| order | 33258 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊(D241⋊C3) |
| unambiguous semidirect product | (C241⋊C3)⋊D23 |
| unambiguous semidirect product | D5543⋊C3 |
| Fitting, socle, derived, Hall | C23×C241 ⋊ C2×C3 |
| Hall | C23×C241⋊C3 ⋊ C2 |
| Hall | D5543 ⋊ C3 |
| Hall | C241⋊C3 ⋊ D23 |
| Sylow | C241 ⋊ C3×D23 |
| Sylow | C23 ⋊ D241⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C23×C241 | C2×C3 |
| LCS | G ▷ C23×C241 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 264 |
| transitive degree | μt(G) = 5543 |
A minimal faithful permutation representation — 2 generators on 264 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,29,30,26,27,31)(28,37,38,32,33,47)(34,50,51,40,41,64)(35,49,45,43,39,36)(42,69,70,53,54,87)(44,67,68,56,52,61)(46,48,66,58,59,62)(55,92,93,72,73,116)(57,90,91,75,71,84)(60,65,89,77,78,85)(63,105,103,81,82,80)(74,121,122,95,96,151)(76,119,120,98,94,113)(79,88,118,100,101,114)(83,134,135,106,107,109)(86,140,132,110,111,104)(97,156,157,124,125,190)(99,154,155,127,123,148)(102,117,153,129,130,149)(108,167,168,136,137,144)(112,174,172,141,142,139)(115,179,165,145,146,133)(126,195,196,159,160,228)(128,193,194,161,158,187)(131,152,192,162,163,188)(138,202,203,169,170,183)(143,207,208,175,176,178)(147,213,205,180,181,173)(150,218,200,184,185,166)(164,191,229,197,198,226)(171,230,233,199,204,222)(177,236,237,209,210,217)(182,240,238,214,215,212)(186,245,234,219,220,206)(189,250,231,223,224,201)(211,254,255,232,227,249)(216,258,259,241,242,244)(221,261,256,246,247,239)(225,262,243,251,252,235)(248,263,264,257,253,260);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,28,34,42,55,74,97,126,161,127,98,75,56,43,27,31,36,46,60,79,102,131,164,199,169,136,106,81,59,33,39,47,61,80,104,133,166,201,232,209,175,141,110,78,41,52,62,64,84,109,139,173,206,235,241,214,180,145,101,54,71,82,85,87,113,144,178,212,239,257,246,219,184,130,73,94,107,111,114,116,148,183,217,244,260,263,251,223,163,96,123,137,142,146,149,151,187,222,249,262,261,258,254,198,125,158,170,176,181,185,188,190,196,229,250,245,240,236,230,160,195,204,210,215,220,224,226,228,157,192,218,213,207,202,193,156,191,227,242,247,252,255,233,194,122,153,179,174,167,154,121,152,189,225,253,264,259,237,203,155,93,118,140,134,119,92,117,150,186,221,248,256,238,208,168,120,70,89,105,90,69,88,115,147,182,216,243,234,205,172,135,91,51,66,67,50,65,86,112,143,177,211,231,200,165,132,103,68,38,49,37,48,63,83,108,138,171,197,162,129,100,77,58,45,30,29,35,44,57,76,99,128,159,124,95,72,53,40,32,26)Polycyclic presentation:
2,3,23,241|c^a=c^22,d^a=d^240,d^b=d^15959 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33258, 12);