Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#13741 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | D32 |
| commutator, Frattini, cosocle | C32 | C23×C11 |
| socle | C2×C11 | C2×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ C2×C11×C32 ◁ 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)(14,27)(15,22)(16,36)(17,43)(18,48)(19,51)(20,26)(21,29)(23,39)(24,38)(25,45)(28,67)(30,75)(31,77)(32,70)(33,79)(34,84)(35,66)(37,63)(40,98)(41,101)(42,88)(44,60)(46,107)(47,104)(49,68)(50,109)(52,69)(53,71)(54,90)(55,89)(56,112)(57,74)(58,82)(59,81)(61,122)(62,97)(64,94)(65,73)(72,129)(76,131)(78,127)(80,119)(83,133)(85,118)(86,92)(87,95)(91,126)(93,103)(96,111)(99,113)(100,138)(102,125)(105,123)(106,121)(108,120)(110,128)(114,130)(115,139)(116,134)(117,136)(124,132)(135,137);(12,14)(13,21)(15,28)(16,37)(17,44)(18,49)(19,52)(20,32)(22,53)(23,60)(24,63)(25,65)(26,57)(27,36)(29,39)(30,38)(31,45)(33,80)(34,85)(35,78)(40,99)(41,102)(42,91)(43,54)(46,108)(47,106)(48,55)(50,110)(51,56)(58,118)(59,119)(61,123)(62,121)(64,126)(66,114)(67,79)(68,98)(69,101)(70,88)(71,82)(72,81)(73,122)(74,97)(75,86)(76,94)(77,87)(83,134)(84,113)(89,92)(90,95)(93,105)(96,112)(100,139)(103,125)(104,116)(107,115)(109,117)(111,120)(124,129)(127,133)(128,138)(130,132)(131,135)(136,137);(12,15,20,19)(13,22,26,51)(14,28,32,52)(16,33,42,41)(17,34,47,46)(18,35,50,45)(21,53,57,56)(23,58,62,111)(24,59,64,103)(25,48,66,109)(27,67,70,69)(29,71,74,112)(30,72,76,105)(31,49,78,110)(36,79,88,101)(37,80,91,102)(38,81,94,93)(39,82,97,96)(40,83,100,95)(43,84,104,107)(44,85,106,108)(54,113,116,115)(55,114,117,65)(60,118,121,120)(61,86,124,135)(63,119,126,125)(68,127,128,77)(73,89,130,136)(75,129,131,123)(87,98,133,138)(90,99,134,139)(92,132,137,122)770 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6729);