Finite Group DatabasebetaFinite Group Databasebeta| order | 30246 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C71 |
| unambiguous semidirect product | (C71⋊D71)⋊C3 |
| Fitting, socle, derived, Sylow | C712 ⋊ C2×C3 |
| Hall | C712⋊C3 ⋊ C2 |
| Hall | C71⋊D71 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C712 | C2×C3 |
| LCS | G ▷ C712 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 213 |
A minimal faithful permutation representation — 2 generators on 213 points:
(1,2,4)(3,19,20,7,8,34)(5,32,25,12,13,58)(6,11,42,16,17,30)(9,54,55,23,24,107)(10,59,48,26,27,21)(14,78,80,36,37,135)(15,71,83,39,40,120)(18,31,89,38,44,68)(22,75,86,51,52,132)(28,114,53,61,62,105)(29,119,102,64,65,50)(33,35,128,41,72,76)(43,112,97,60,87,154)(45,69,150,91,92,125)(46,47,122,94,84,67)(49,70,139,98,99,116)(56,74,111,103,66,131)(57,160,144,101,110,177)(63,133,73,77,117,82)(79,81,191,85,137,140)(88,171,106,115,145,164)(90,170,157,113,147,153)(93,126,199,151,152,183)(95,96,180,100,136,124)(104,166,123,108,109,179)(118,208,198,174,175,192)(121,195,162,143,169,161)(127,181,167,168,173,163)(129,193,158,159,185,200)(130,134,196,149,146,188)(138,141,209,142,184,194)(148,207,165,172,197,204)(155,156,211,187,189,182)(176,212,203,186,210,205)(178,213,202,190,206,201);(1,3,9,25,59,111,80,139,114,68,124,181,133,170,206,194,208,207,183,196,212,211,150,145,185,191,195,87,128,179,122,42,86,144,83,119,65,120,177,132,30,67,123,76,112,169,140,193,171,125,182,210,188,199,197,175,209,213,147,117,173,180,89,62,116,135,131,27,58,23,7)(2,5,14,38,82,142,151,155,158,97,46,57,102,51,104,162,164,203,165,201,167,53,103,54,8,21,49,100,153,192,130,45,79,33,6,15,39,16,41,85,91,149,198,157,95,98,48,19,24,56,105,163,202,204,205,106,161,108,22,50,101,94,154,200,187,93,138,73,18,36,12)(4,10,28,63,118,176,159,35,75,64,17,43,88,146,141,168,99,13,34,74,44,90,148,156,81,109,160,40,84,143,92,126,178,96,37,55,107,78,136,190,152,69,121,47,71,110,166,137,189,172,113,31,66,20,32,70,127,184,134,115,60,11,29,52,72,129,186,174,77,61,26)Polycyclic presentation:
2,3,71^2|c^a=c^70,d^a=d^70,c^b=c^67d^5,d^b=c^40d^3846 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30246, 7);