Finite Group DatabasebetaFinite Group Databasebeta| order | 24450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C25⋊(D163⋊C3) |
| unambiguous semidirect product | (C163⋊C3)⋊D25 |
| unambiguous semidirect product | D4075⋊C3 |
| Fitting, derived, Hall | C25×C163 ⋊ C2×C3 |
| Hall | C25×C163⋊C3 ⋊ C2 |
| Hall | D4075 ⋊ C3 |
| Hall | C163⋊C3 ⋊ D25 |
| Sylow | C163 ⋊ C3×D25 |
| Sylow | C25 ⋊ D163⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C25×C163 | C2×C3 |
| Frattini | C5 | C5⋊(D163⋊C3) |
| socle | C5×C163 | C3×D5 |
| LCS | G ▷ C25×C163 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 188 |
| transitive degree | μt(G) = 4075 |
A minimal faithful permutation representation — 2 generators on 188 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)(51,98,99,72,73,127)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,132,133,101,102,154)(76,130,131,104,100,124)(79,94,129,106,107,125)(85,139,140,113,114,121)(89,144,103,118,119,116)(92,149,138,122,123,110)(105,155,156,135,134,152)(108,128,151,137,126,153)(115,157,158,141,142,150)(120,160,136,145,146,148)(143,161,162,159,147,163)(165,167)(166,170)(168,173)(169,176)(171,179)(172,178)(174,175)(177,183)(180,184)(181,185)(182,187)(186,188);(1,2,5,11,19,32,51,74,103,112,68,28,43,44,27,42,63,89,120,147,162,156,99,129,149,144,139,130,98,128,107,50,71,84,88,91,93,124,150,159,141,113,83,58,36,10,16,24,38,57,81,110,137,106,77,54,35,22,7,6,12,21,34,53,76,105,136,140,97,47,66,82,67,46,65,92,102,134,142,146,133,151,153,154,160,157,155,132,122,78,31,48,59,62,64,90,121,148,135,104,75,52,33,20,4,8,13,23,37,56,79,108,138,109,80,45,15,26,14,25,40,60,85,115,143,158,131,70,95,117,111,96,69,94,126,73,100,114,119,123,125,127,152,163,161,145,118,87,55,18,29,39,41,61,86,116,101,72,49,30,17,9,3)(164,178,188,181,167,170,184,182,173,176,179,177,175,174,183,171,169,168,187,180,166,165,185,186,172)Polycyclic presentation:
2,3,5,163,5|c^5=e^4,c^a=c^4e,d^a=d^162,e^a=e^4,d^b=d^104717 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24450, 12);