Finite Group DatabasebetaFinite Group Databasebeta| order | 33666 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C181⋊C3)⋊D31 |
| Fitting, socle, derived, Hall | C31×C181 ⋊ C2×C3 |
| Hall | C31×C181⋊C3 ⋊ C2 |
| Hall | D5611 ⋊ C3 |
| Hall | C181⋊C3 ⋊ D31 |
| Sylow | C181 ⋊ C3×D31 |
| Sylow | C31 ⋊ D181⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C31×C181 | C2×C3 |
| LCS | G ▷ C31×C181 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 212 |
| transitive degree | μt(G) = 5611 |
A minimal faithful permutation representation — 2 generators on 212 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)(30,31)(33,37,38,34,35,39)(36,45,46,40,41,55)(42,58,59,48,49,72)(43,57,53,51,47,44)(50,77,78,61,62,95)(52,75,76,64,60,69)(54,56,74,66,67,70)(63,100,101,80,81,124)(65,98,99,83,79,92)(68,73,97,85,86,93)(71,113,111,89,90,88)(82,129,130,103,104,159)(84,127,128,106,102,121)(87,96,126,108,109,122)(91,142,143,114,115,117)(94,148,140,118,119,112)(105,164,165,132,133,193)(107,162,163,135,131,156)(110,125,161,137,138,157)(116,175,176,144,145,152)(120,181,179,149,150,147)(123,184,173,153,154,141)(134,185,180,155,167,204)(136,196,197,168,166,190)(139,160,195,170,171,191)(146,202,203,177,178,186)(151,198,206,169,182,183)(158,207,200,187,188,174)(172,194,205,189,199,211)(192,212,210,208,209,201);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(32,33,36,42,50,63,82,105,134,153,109,62,79,90,93,95,121,152,183,165,195,207,167,198,202,196,164,194,209,211,204,179,143,99,59,74,75,58,73,94,120,151,168,135,106,83,64,51,35,39,44,54,68,87,110,139,172,187,138,81,102,115,119,122,124,156,186,197,130,161,184,181,175,162,129,160,192,210,200,173,140,111,76,46,57,45,56,71,91,116,146,177,144,114,89,67,41,47,55,69,88,112,141,174,201,208,171,104,131,145,150,154,157,159,190,203,163,101,126,148,142,127,100,125,158,189,170,137,108,85,66,53,38,37,43,52,65,84,107,136,169,149,118,86,49,60,70,72,92,117,147,180,205,212,199,133,166,178,182,185,188,191,193,206,176,128,78,97,113,98,77,96,123,155,132,103,80,61,48,40,34)Polycyclic presentation:
2,3,31,181|c^a=c^30,d^a=d^180,d^b=d^48981 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33666, 21);