Finite Group DatabasebetaFinite Group Databasebeta| order | 31494 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C29⋊(D181⋊C3) |
| unambiguous semidirect product | (C181⋊C3)⋊D29 |
| unambiguous semidirect product | D5249⋊C3 |
| Fitting, socle, derived, Hall | C29×C181 ⋊ C2×C3 |
| Hall | C29×C181⋊C3 ⋊ C2 |
| Hall | D5249 ⋊ C3 |
| Hall | C181⋊C3 ⋊ D29 |
| Sylow | C181 ⋊ C3×D29 |
| Sylow | C29 ⋊ D181⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C29×C181 | C2×C3 |
| LCS | G ▷ C29×C181 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 210 |
| transitive degree | μt(G) = 5249 |
A minimal faithful permutation representation — 2 generators on 210 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(31,35,36,32,33,37)(34,43,44,38,39,53)(40,56,57,46,47,70)(41,55,51,49,45,42)(48,75,76,59,60,93)(50,73,74,62,58,67)(52,54,72,64,65,68)(61,98,99,78,79,122)(63,96,97,81,77,90)(66,71,95,83,84,91)(69,111,109,87,88,86)(80,127,128,101,102,157)(82,125,126,104,100,119)(85,94,124,106,107,120)(89,140,141,112,113,115)(92,146,138,116,117,110)(103,162,163,130,131,191)(105,160,161,133,129,154)(108,123,159,135,136,155)(114,173,174,142,143,150)(118,179,177,147,148,145)(121,182,171,151,152,139)(132,183,178,153,165,202)(134,194,195,166,164,188)(137,158,193,168,169,189)(144,200,201,175,176,184)(149,196,204,167,180,181)(156,205,198,185,186,172)(170,192,203,187,197,209)(190,210,208,206,207,199);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,31,34,40,48,61,80,103,132,151,107,60,77,88,91,93,119,150,181,163,193,205,165,196,200,194,162,192,207,209,202,177,141,97,57,72,73,56,71,92,118,149,166,133,104,81,62,49,33,37,42,52,66,85,108,137,170,185,136,79,100,113,117,120,122,154,184,195,128,159,182,179,173,160,127,158,190,208,198,171,138,109,74,44,55,43,54,69,89,114,144,175,142,112,87,65,39,45,53,67,86,110,139,172,199,206,169,102,129,143,148,152,155,157,188,201,161,99,124,146,140,125,98,123,156,187,168,135,106,83,64,51,36,35,41,50,63,82,105,134,167,147,116,84,47,58,68,70,90,115,145,178,203,210,197,131,164,176,180,183,186,189,191,204,174,126,76,95,111,96,75,94,121,153,130,101,78,59,46,38,32)Polycyclic presentation:
2,3,29,181|c^a=c^28,d^a=d^180,d^b=d^48918 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31494, 12);