Finite Group DatabasebetaFinite Group Databasebeta| order | 24318 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22549226 |
| Fitting, socle, derived | C3×C7×C193 ⋊ C2×C3 |
| Hall | C3×#16860995 ⋊ C2 |
| Hall | C7×C193 ⋊ C3×D3 |
| Sylow | C193 ⋊ D3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊(D193⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C193 | C2×C3 |
| LCS | G ▷ C3×C7×C193 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 203 |
| transitive degree | μt(G) = 4053 |
A minimal faithful permutation representation — 2 generators on 203 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,168)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,145,146,113,114,121)(89,152,150,118,119,116)(92,157,143,122,123,110)(103,172,142,136,137,174)(105,170,171,138,135,165)(108,129,169,140,141,166)(115,177,178,147,148,161)(120,182,183,153,154,156)(124,186,180,158,159,151)(127,189,175,162,163,144)(139,192,176,167,173,193)(149,190,181,164,179,191)(155,187,185,160,184,188)(195,196,197)(198,200,199)(202,203);(1,2,5,11,19,32,51,74,103,102,135,148,154,159,163,166,168,193,175,143,109,80,45,15,26,14,25,40,60,85,115,149,162,107,50,71,84,88,91,93,125,161,188,183,146,97,47,66,82,67,46,65,92,124,160,153,118,87,55,18,29,39,41,61,86,116,151,181,171,99,130,157,152,145,131,98,129,167,138,104,75,52,33,20,4,8,13,23,37,56,79,108,142,137,173,179,184,187,190,192,172,174,140,106,77,54,35,22,7,6,12,21,34,53,76,105,139,141,73,100,114,119,123,126,128,165,191,180,150,112,68,28,43,44,27,42,63,89,120,155,158,122,78,31,48,59,62,64,90,121,156,185,178,132,70,95,117,111,96,69,94,127,164,147,113,83,58,36,10,16,24,38,57,81,110,144,176,134,169,189,186,182,177,170,133,136,101,72,49,30,17,9,3)(194,195,196,198,197,199,200)(201,202,203)Polycyclic presentation:
2,3^2,7,193|c^a=c^2,e^a=e^192,d^b=d^2,e^b=e^84687 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24318, 51);