Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C19×C32 · Schur multiplier (H2) = C2 |
| direct product | C19×#7514 |
| subgroup | G/H | |
|---|---|---|
| center | C19×C32 | D4 |
| commutator | C4 | C2×C19×C32 |
| Frattini, cosocle | C2×C32 | C22×C19 |
| socle | C2×C19 | C22⋊C32 |
| prime radical | #41×C19 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C19×C32 ◁ C2×C19×C32 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 2 generators on 147 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,31,25,32,47,66,26,33,48,67,51,70,94,115,27,34,49,68,52,71,95,116,54,73,97,118,99,120,135,143,28,35,50,69,53,72,96,117,55,74,98,119,100,121,136,144,23,30,43,62,44,63,88,110,45,64,89,111,90,112,131,141)(22,36,38,75,39,76,81,122,40,77,82,123,83,124,128,145,37,60,78,65,79,106,125,113,80,107,126,114,127,139,146,142,41,61,46,91,84,108,92,132,85,109,93,133,129,140,134,147,42,29,56,57,86,58,101,103,87,59,102,104,130,105,137,138);(20,22)(21,29)(23,42)(24,46)(25,39)(26,40)(27,37)(28,41)(30,61)(31,65)(32,58)(33,59)(34,36)(35,60)(38,50)(43,78)(44,86)(45,87)(47,92)(48,93)(49,56)(51,83)(52,79)(53,84)(54,80)(55,85)(57,69)(62,75)(63,108)(64,109)(66,113)(67,114)(68,91)(70,105)(71,76)(72,106)(73,77)(74,107)(81,96)(82,98)(88,125)(89,126)(90,130)(94,134)(95,101)(97,102)(99,127)(100,129)(103,117)(104,119)(110,122)(111,123)(112,140)(115,142)(116,132)(118,133)(120,124)(121,139)(128,136)(131,146)(135,137)(138,144)(141,145)(143,147)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 502);