Finite Group DatabasebetaFinite Group Databasebeta| order | 24450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, derived, Hall | C52×C163 ⋊ C2×C3 |
| Hall | C52⋊(C163⋊C3) ⋊ C2 |
| Hall | C5⋊D815 ⋊ C3 |
| Sylow | C163 ⋊ (C5⋊D5)⋊C3 |
| Sylow | C52 ⋊ D163⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C163 | C2×C3 |
| LCS | G ▷ C52×C163 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 178 |
| transitive degree | μt(G) = 2445 |
A minimal faithful permutation representation — 2 generators on 178 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)(164,165,167)(166,176,172,169,170,178)(168,173,177,174,175,171);(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)(164,166,171,177,169)(165,168,170,176,174)(167,172,173,175,178)Polycyclic presentation:
2,3,5^2,163|c^a=c^4,d^a=d^4,e^a=e^162,c^b=cd,d^b=c^2d^3,e^b=e^104685 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24450, 19);