Finite Group DatabasebetaFinite Group Databasebeta| order | 24978 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊(D181⋊C3) |
| unambiguous semidirect product | (C181⋊C3)⋊D23 |
| unambiguous semidirect product | D4163⋊C3 |
| Fitting, socle, derived, Hall | C23×C181 ⋊ C2×C3 |
| Hall | C23×C181⋊C3 ⋊ C2 |
| Hall | D4163 ⋊ C3 |
| Hall | C181⋊C3 ⋊ D23 |
| Sylow | C181 ⋊ C3×D23 |
| Sylow | C23 ⋊ D181⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C23×C181 | C2×C3 |
| LCS | G ▷ C23×C181 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 204 |
| transitive degree | μt(G) = 4163 |
A minimal faithful permutation representation — 2 generators on 204 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)(74,133,134,101,102,162)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,144,145,113,114,121)(89,150,148,118,119,116)(92,153,142,122,123,110)(103,154,149,124,136,173)(105,165,166,137,135,159)(108,129,164,139,140,160)(115,171,172,146,147,155)(120,167,175,138,151,152)(127,176,169,156,157,143)(141,163,174,158,168,180)(161,181,179,177,178,170)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204);(1,2,5,11,19,32,51,74,103,122,78,31,48,59,62,64,90,121,152,134,164,176,136,167,171,165,133,163,178,180,173,148,112,68,28,43,44,27,42,63,89,120,137,104,75,52,33,20,4,8,13,23,37,56,79,108,141,156,107,50,71,84,88,91,93,125,155,166,99,130,153,150,144,131,98,129,161,179,169,142,109,80,45,15,26,14,25,40,60,85,115,146,113,83,58,36,10,16,24,38,57,81,110,143,170,177,140,73,100,114,119,123,126,128,159,172,132,70,95,117,111,96,69,94,127,158,139,106,77,54,35,22,7,6,12,21,34,53,76,105,138,118,87,55,18,29,39,41,61,86,116,149,174,181,168,102,135,147,151,154,157,160,162,175,145,97,47,66,82,67,46,65,92,124,101,72,49,30,17,9,3)(182,183,185,187,189,191,193,195,197,199,201,203,204,202,200,198,196,194,192,190,188,186,184)Polycyclic presentation:
2,3,23,181|c^a=c^22,d^a=d^180,d^b=d^48729 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24978, 12);