Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11×C32 · Schur multiplier (H2) = C22 |
| direct product | C11×#13737 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C64 | C22 |
| commutator | C2 | C22×C11×C32 |
| Frattini, cosocle | C32 | C23×C11 |
| socle | C2×C11 | C22×C32 |
| prime radical | C11×#41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C11×C64 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,21,16,22,31,46,17,23,32,47,35,50,66,86,18,24,33,48,36,51,67,87,38,53,69,89,72,92,106,121,19,25,34,49,37,52,68,88,39,54,70,90,73,93,107,122,40,55,71,91,74,94,108,123,75,95,109,124,110,125,131,137)(14,20,27,42,28,43,60,80,29,44,61,81,63,83,102,117,30,45,62,82,64,84,103,118,65,85,104,119,105,120,130,136,26,41,56,76,57,77,96,111,58,78,97,112,99,114,126,132,59,79,98,113,100,115,127,133,101,116,128,134,129,135,138,139);(12,14)(13,20)(15,27)(16,28)(17,29)(18,30)(19,26)(21,42)(22,43)(23,44)(24,45)(25,41)(31,60)(32,61)(33,62)(34,56)(35,63)(36,64)(37,57)(38,65)(39,58)(40,59)(46,80)(47,81)(48,82)(49,76)(50,83)(51,84)(52,77)(53,85)(54,78)(55,79)(66,102)(67,103)(68,96)(69,104)(70,97)(71,98)(72,105)(73,99)(74,100)(75,101)(86,117)(87,118)(88,111)(89,119)(90,112)(91,113)(92,120)(93,114)(94,115)(95,116)(106,130)(107,126)(108,127)(109,128)(110,129)(121,136)(122,132)(123,133)(124,134)(125,135)(131,138)(137,139);(14,26)(20,41)(27,56)(28,57)(29,58)(30,59)(42,76)(43,77)(44,78)(45,79)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(102,126)(103,127)(104,128)(105,129)(117,132)(118,133)(119,134)(120,135)(130,138)(136,139)1760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6725);