Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C23×C19 |
| prime radical | C19×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C8×C19 ◁ C2×C16×C19 ◁ C2×C19×C32 ◁ G | |
126 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 58 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)(22,34)(23,40)(24,47)(25,52)(26,56)(27,29)(28,35)(30,43)(31,42)(32,49)(33,54)(36,80)(37,84)(38,87)(39,73)(41,68)(44,100)(45,104)(46,72)(48,65)(50,112)(51,106)(53,74)(55,114)(57,75)(58,79)(59,92)(60,91)(61,117)(62,76)(63,81)(64,85)(66,126)(67,128)(69,132)(70,77)(71,78)(82,143)(83,142)(86,144)(88,95)(89,98)(90,102)(93,133)(94,123)(96,108)(97,107)(99,116)(101,130)(103,137)(105,131)(109,127)(110,129)(111,121)(113,125)(115,136)(118,146)(119,145)(120,147)(122,140)(124,141)(134,139)(135,138);(20,22,27,39)(21,28,29,58)(23,41,46,94)(24,48,51,111)(25,53,55,115)(26,57,54,38)(30,65,62,121)(31,68,63,123)(32,70,64,124)(33,71,56,61)(34,72,73,40)(35,76,79,43)(36,81,83,42)(37,85,86,49)(44,101,103,147)(45,105,102,93)(47,59,106,118)(50,113,96,110)(52,60,114,119)(66,127,122,143)(67,129,116,125)(69,133,107,131)(74,137,136,100)(75,90,87,104)(77,140,141,126)(78,99,117,128)(80,134,142,88)(82,97,109,132)(84,135,144,89)(91,139,145,95)(92,138,146,98)(108,130,112,120);(21,29)(28,58)(30,62)(31,63)(32,64)(33,56)(34,73)(40,72)(47,106)(52,114)(59,118)(60,119)(61,71)(65,121)(66,122)(67,116)(68,123)(69,107)(70,124)(74,136)(75,87)(80,142)(84,144)(88,134)(89,135)(90,104)(100,137)(108,112)(120,130)(125,129)(127,143)(131,133)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6731);