Finite Group DatabasebetaFinite Group Databasebeta| order | 30366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#23778288 |
| Fitting, socle, derived | C3×C7×C241 ⋊ C2×C3 |
| Hall | C3×#19099464 ⋊ 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)(113,181,182,146,147,215)(115,179,180,149,145,174)(118,139,178,148,151,175)(125,188,189,156,157,170)(130,195,196,162,163,165)(134,201,193,167,168,160)(137,206,186,171,172,153)(150,216,217,184,183,214)(158,218,219,190,191,210)(164,222,223,197,198,205)(169,229,227,202,203,200)(173,234,221,207,208,194)(176,239,185,211,212,187)(192,240,241,220,213,242)(199,243,244,224,225,238)(204,245,246,230,231,233)(209,248,226,235,236,228)(232,249,250,247,237,251);(1,2,3)(4,5,7,10,9,8,6)(11,12,15,21,29,42,61,84,113,148,116,87,64,45,32,17,16,22,31,44,63,86,115,150,185,186,152,119,90,55,25,36,24,35,50,70,95,125,158,192,221,193,159,122,78,38,53,54,37,52,73,99,130,164,199,226,227,196,155,107,57,76,92,77,56,75,102,134,169,204,232,246,223,189,142,80,105,127,121,106,79,104,137,173,209,237,250,244,219,180,109,140,166,161,154,141,108,139,176,213,225,231,236,241,217,144,178,206,201,195,188,179,143,147,183,191,198,203,208,212,182,215,239,234,229,222,218,216,181,112,145,157,163,168,172,175,177,214,242,248,245,243,240,211,151,83,110,124,129,133,136,138,174,210,238,251,249,235,207,171,117,60,81,94,98,101,103,135,170,205,233,247,230,202,167,132,88,41,58,69,72,74,100,131,165,200,228,224,197,162,128,97,65,28,39,49,51,71,96,126,160,194,220,190,156,123,93,68,46,20,26,34,48,67,91,120,153,187,184,149,114,85,62,43,30,14,18,23,33,47,66,89,118,146,111,82,59,40,27,19,13)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^225852 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30366, 45);