Finite Group DatabasebetaFinite Group Databasebeta| order | 31578 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C277⋊C3)⋊D19 |
| Fitting, socle, derived, Hall | C19×C277 ⋊ C2×C3 |
| Hall | C19×C277⋊C3 ⋊ C2 |
| Hall | D5263 ⋊ C3 |
| Hall | C277⋊C3 ⋊ D19 |
| Sylow | C277 ⋊ C3×D19 |
| Sylow | C19 ⋊ D277⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C19×C277 | C2×C3 |
| LCS | G ▷ C19×C277 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 296 |
| transitive degree | μt(G) = 5263 |
A minimal faithful permutation representation — 2 generators on 296 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,25,26,22,23,27)(24,33,34,28,29,43)(30,46,47,36,37,60)(31,45,41,39,35,32)(38,65,66,49,50,83)(40,63,64,52,48,57)(42,44,62,54,55,58)(51,88,89,68,69,112)(53,86,87,71,67,80)(56,61,85,73,74,81)(59,101,99,77,78,76)(70,117,118,91,92,147)(72,115,116,94,90,109)(75,84,114,96,97,110)(79,130,131,102,103,105)(82,136,128,106,107,100)(93,152,153,120,121,188)(95,150,151,123,119,144)(98,113,149,125,126,145)(104,165,166,132,133,140)(108,172,170,137,138,135)(111,177,163,141,142,129)(122,193,194,155,156,235)(124,191,192,158,154,185)(127,148,190,160,161,186)(134,206,207,167,168,181)(139,213,214,173,174,176)(143,219,211,178,179,171)(146,224,204,182,183,164)(157,240,241,196,197,279)(159,238,239,199,195,232)(162,189,237,201,202,233)(169,251,252,208,209,228)(175,258,259,215,216,223)(180,264,262,220,221,218)(184,268,256,225,226,212)(187,273,249,229,230,205)(198,253,254,243,210,286)(200,281,282,244,242,277)(203,236,280,246,247,278)(217,291,248,260,261,272)(222,292,285,265,266,267)(227,293,290,269,270,263)(231,295,289,274,275,257)(234,294,284,271,276,250)(245,287,288,283,255,296);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,24,30,38,51,70,93,122,157,198,239,153,190,224,219,213,206,191,152,189,234,269,225,182,126,69,90,103,107,110,112,144,181,223,267,284,249,204,163,128,99,64,34,45,33,44,59,79,104,134,169,210,255,288,241,280,294,295,293,292,291,286,281,240,260,215,173,137,106,74,37,48,58,60,80,105,135,171,212,257,290,262,214,166,116,66,85,101,86,65,84,111,143,180,222,247,156,195,209,216,221,226,230,233,235,277,283,244,199,158,123,94,71,52,39,23,27,32,42,56,75,98,127,162,203,248,252,192,118,149,177,172,165,150,117,148,187,231,274,229,161,92,119,133,138,142,145,147,185,228,272,246,201,160,125,96,73,54,41,26,25,31,40,53,72,95,124,159,200,245,282,194,237,273,268,264,258,251,238,193,236,265,220,178,141,97,50,67,78,81,83,109,140,176,218,263,289,256,211,170,131,87,47,62,63,46,61,82,108,139,175,217,197,242,254,261,266,270,275,276,278,279,296,287,253,208,167,132,102,77,55,29,35,43,57,76,100,129,164,205,250,285,259,207,151,89,114,136,130,115,88,113,146,184,227,271,202,121,154,168,174,179,183,186,188,232,243,196,155,120,91,68,49,36,28,22)Polycyclic presentation:
2,3,19,277|c^a=c^18,d^a=d^276,d^b=d^116907 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31578, 21);