Finite Group DatabasebetaFinite Group Databasebeta| order | 30366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#23778287 |
| Fitting, socle, derived | C3×C7×C241 ⋊ C2×C3 |
| Hall | C3×#19099463 ⋊ C2 |
| Hall | C7×C241 ⋊ C3×D3 |
| Sylow | C241 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3⋊(D241⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C241 | C2×C3 |
| LCS | G ▷ C3×C7×C241 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 251 |
| transitive degree | μt(G) = 5061 |
A minimal faithful permutation representation — 2 generators on 251 points:
(2,3)(5,8,9,6,7,10)(12,16,17,13,14,18)(15,24,25,19,20,34)(21,37,38,27,28,51)(22,36,32,30,26,23)(29,56,57,40,41,74)(31,54,55,43,39,48)(33,35,53,45,46,49)(42,79,80,59,60,103)(44,77,78,62,58,71)(47,52,76,64,65,72)(50,92,90,68,69,67)(61,108,109,82,83,138)(63,106,107,85,81,100)(66,75,105,87,88,101)(70,121,122,93,94,96)(73,127,119,97,98,91)(84,143,144,111,112,177)(86,141,142,114,110,135)(89,104,140,116,117,136)(95,154,155,123,124,131)(99,161,159,128,129,126)(102,166,152,132,133,120)(113,182,183,146,147,215)(115,180,181,148,145,174)(118,139,179,149,150,175)(125,189,190,156,157,170)(130,194,195,162,163,165)(134,200,192,167,168,160)(137,205,187,171,172,153)(151,178,216,184,185,213)(158,217,220,186,191,209)(164,223,224,196,197,204)(169,227,225,201,202,199)(173,232,221,206,207,193)(176,237,218,210,211,188)(198,241,242,219,214,236)(203,245,246,228,229,231)(208,248,243,233,234,226)(212,249,230,238,239,222)(235,250,251,244,240,247);(1,2,3)(4,5,7,10,9,8,6)(11,12,15,21,29,42,61,84,113,148,114,85,62,43,30,14,18,23,33,47,66,89,118,151,186,156,123,93,68,46,20,26,34,48,67,91,120,153,188,219,196,162,128,97,65,28,39,49,51,71,96,126,160,193,222,228,201,167,132,88,41,58,69,72,74,100,131,165,199,226,244,233,206,171,117,60,81,94,98,101,103,135,170,204,231,247,250,238,210,150,83,110,124,129,133,136,138,174,209,236,249,248,245,241,185,112,145,157,163,168,172,175,177,183,216,237,232,227,223,217,147,182,191,197,202,207,211,213,215,144,179,205,200,194,189,180,143,178,214,229,234,239,242,220,181,109,140,166,161,154,141,108,139,176,212,240,251,246,224,190,142,80,105,127,121,106,79,104,137,173,208,235,243,225,195,155,107,57,76,92,77,56,75,102,134,169,203,230,221,192,159,122,78,38,53,54,37,52,73,99,130,164,198,218,187,152,119,90,55,25,36,24,35,50,70,95,125,158,184,149,116,87,64,45,32,17,16,22,31,44,63,86,115,146,111,82,59,40,27,19,13)Polycyclic presentation:
2,3^2,7,241|c^a=c^2,d^a=d^6,e^a=e^240,d^b=d^2,e^b=e^15852 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30366, 46);