Finite Group DatabasebetaFinite Group Databasebeta| order | 30366 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#23778284 |
| 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 ⋊ D3×C241⋊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)(12,13,15)(14,17,22)(16,20,19)(18,24,32)(21,27,29)(23,30,26)(25,34,45)(28,37,42)(31,40,39)(33,43,36)(35,47,61)(38,50,58)(41,53,55)(44,56,52)(46,59,49)(48,63,80)(51,66,77)(54,69,74)(57,72,71)(60,75,68)(62,78,65)(64,82,102)(67,85,99)(70,88,96)(73,91,93)(76,94,90)(79,97,87)(81,100,84)(83,104,127)(86,107,124)(89,110,121)(92,113,118)(95,116,115)(98,119,112)(101,122,109)(103,125,106)(105,129,155)(108,132,152)(111,135,149)(114,138,146)(117,141,143)(120,144,140)(123,147,137)(126,150,134)(128,153,131)(130,157,175)(133,159,182)(136,162,179)(139,165,176)(142,168,173)(145,171,170)(148,174,167)(151,177,164)(154,180,161)(156,183,158)(160,187,200)(163,189,207)(166,192,204)(169,195,201)(172,197,199)(178,202,194)(181,205,191)(184,208,188)(185,203,186)(190,212,221)(193,214,226)(196,217,223)(198,219,222)(206,224,216)(209,227,213)(210,225,211)(215,231,237)(218,233,239)(220,235,238)(228,240,232)(229,241,230)(234,244,247)(236,245,248)(242,249,243)(246,250,251);(1,2,3)(4,5,7,10,9,8,6)(11,12,14,18,25,35,48,64,83,105,130,144,94,56,30,15,19,26,36,49,65,84,106,131,158,186,174,119,75,43,22,29,39,52,68,87,109,134,161,188,211,202,147,97,59,32,42,55,71,90,112,137,164,191,213,230,224,177,122,78,45,58,74,93,115,140,167,194,216,232,243,240,205,150,100,61,77,96,118,143,170,157,185,210,229,242,249,227,180,125,80,99,121,146,173,199,187,129,156,184,209,228,241,208,153,102,124,149,176,201,222,212,159,104,128,154,181,206,225,183,127,152,179,204,223,238,231,189,132,82,103,126,151,178,203,155,182,207,226,239,248,244,214,162,107,63,81,101,123,148,175,200,221,237,247,251,250,233,192,135,85,47,62,79,98,120,145,172,198,220,236,246,245,217,165,110,66,34,46,60,76,95,117,142,169,196,218,234,235,195,138,88,50,24,33,44,57,73,92,114,139,166,193,215,219,168,113,69,37,17,23,31,41,54,70,89,111,136,163,190,197,141,91,53,27,13,16,21,28,38,51,67,86,108,133,160,171,116,72,40,20)Polycyclic presentation:
2,3^2,7,241|c^a=c^2,d^a=d^6,d^b=d^2,e^b=e^225972 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30366, 47);