Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C19 |
| prime radical | #41×C19 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C19×#131 ◁ C19×#504 ◁ G | |
180 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,28,35)(22,36,43,79)(23,44,48,71)(24,49,53,108)(25,34,27,32)(26,56,57,72)(29,58,63,39)(30,54,66,46)(31,67,70,37)(33,55,73,50)(38,84,88,128)(40,78,42,76)(41,94,95,129)(45,99,102,142)(47,103,104,65)(51,107,52,106)(59,113,115,137)(60,90,62,91)(61,118,119,132)(64,120,121,133)(68,82,69,83)(74,92,127,86)(75,112,117,80)(77,93,130,89)(81,97,122,123)(85,96,111,116)(87,138,139,126)(98,110,124,100)(101,105,109,141)(114,131,134,143)(125,144,145,147)(135,140,146,136);(20,22,37,80,26,41,51,90,25,40,82,118,55,93,53,58,28,43,67,112,57,95,52,91,27,42,83,132,50,89,24,39)(21,29,49,77,33,61,68,76,32,60,106,129,72,117,70,79,35,63,108,130,73,119,69,78,34,62,107,94,56,75,31,36)(23,38,81,131,47,87,109,113,54,92,120,144,110,140,102,111,48,88,122,143,104,139,101,137,46,86,133,147,100,136,45,85)(30,59,105,126,65,114,123,128,71,116,142,146,124,145,121,127,66,115,141,138,103,134,97,84,44,96,99,135,98,125,64,74);(20,23)(21,30)(22,38)(24,45)(25,54)(26,47)(27,46)(28,48)(29,59)(31,64)(32,71)(33,65)(34,44)(35,66)(36,74)(37,81)(39,85)(40,92)(41,87)(42,86)(43,88)(49,105)(50,100)(51,109)(52,101)(53,102)(55,110)(56,98)(57,104)(58,111)(60,116)(61,114)(62,96)(63,115)(67,122)(68,123)(69,97)(70,121)(72,124)(73,103)(75,125)(76,128)(77,126)(78,84)(79,127)(80,131)(82,120)(83,133)(89,136)(90,113)(91,137)(93,140)(94,135)(95,139)(99,107)(106,142)(108,141)(112,143)(117,145)(118,144)(119,134)(129,146)(130,138)(132,147)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6689);