Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C19 |
| prime radical | C19×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C19×#131 ◁ C19×#504 ◁ G | |
274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 74 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)(20,21,27,33)(22,34,40,78)(23,41,46,66)(24,47,52,106)(25,53,54,31)(26,55,57,113)(28,58,63,36)(29,45,67,43)(30,68,71,51)(32,72,74,49)(35,79,83,133)(37,77,39,76)(38,88,89,130)(42,93,99,141)(44,65,102,127)(48,73,109,120)(50,111,112,104)(56,69,87,131)(59,116,118,136)(60,84,62,85)(61,107,121,108)(64,122,125,144)(70,110,105,129)(75,86,132,81)(80,90,115,119)(82,137,138,123)(91,128,124,97)(92,100,114,95)(94,134,143,117)(96,103,98,140)(101,135,139,147)(126,145,146,142);(20,22)(21,28)(23,35)(24,48)(25,37)(26,56)(27,40)(29,59)(30,69)(31,60)(32,73)(33,63)(34,47)(36,51)(38,49)(39,54)(41,90)(42,94)(43,81)(44,101)(45,86)(46,83)(50,107)(52,109)(53,62)(55,61)(57,87)(58,68)(64,123)(65,126)(66,119)(67,118)(70,130)(71,131)(72,89)(74,120)(75,103)(76,104)(77,111)(78,106)(79,93)(80,97)(82,95)(84,110)(85,129)(88,105)(91,135)(92,134)(96,142)(98,145)(99,143)(100,138)(102,139)(108,112)(113,121)(114,117)(115,128)(116,122)(124,147)(125,137)(127,146)(132,140)(133,141)(136,144);(20,23,27,46)(21,29,33,67)(22,35,40,83)(24,42,52,99)(25,45,54,43)(26,44,57,102)(28,59,63,118)(30,64,71,125)(31,66,53,41)(32,65,74,127)(34,75,78,132)(36,80,58,115)(37,86,39,81)(38,82,89,138)(47,103,106,140)(48,94,109,143)(49,95,72,100)(50,98,112,96)(51,97,68,128)(55,114,113,92)(56,101,87,139)(60,119,62,90)(61,117,121,134)(69,123,131,137)(70,124,105,91)(73,126,120,146)(76,133,77,79)(84,116,85,136)(88,135,130,147)(93,104,141,111)(107,145,108,142)(110,122,129,144)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6660);