Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13741 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | D32 |
| commutator, Frattini, cosocle | C32 | C23×C13 |
| socle | C2×C13 | C2×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C8×C13 ◁ C2×C13×C16 ◁ C2×C13×C32 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,29)(17,24)(18,38)(19,45)(20,50)(21,53)(22,28)(23,31)(25,41)(26,40)(27,47)(30,69)(32,77)(33,79)(34,72)(35,81)(36,86)(37,68)(39,65)(42,100)(43,103)(44,90)(46,62)(48,109)(49,106)(51,70)(52,111)(54,71)(55,73)(56,92)(57,91)(58,114)(59,76)(60,84)(61,83)(63,124)(64,99)(66,96)(67,75)(74,131)(78,133)(80,129)(82,121)(85,135)(87,120)(88,94)(89,97)(93,128)(95,105)(98,113)(101,115)(102,140)(104,127)(107,125)(108,123)(110,122)(112,130)(116,132)(117,141)(118,136)(119,138)(126,134)(137,139);(14,16)(15,23)(17,30)(18,39)(19,46)(20,51)(21,54)(22,34)(24,55)(25,62)(26,65)(27,67)(28,59)(29,38)(31,41)(32,40)(33,47)(35,82)(36,87)(37,80)(42,101)(43,104)(44,93)(45,56)(48,110)(49,108)(50,57)(52,112)(53,58)(60,120)(61,121)(63,125)(64,123)(66,128)(68,116)(69,81)(70,100)(71,103)(72,90)(73,84)(74,83)(75,124)(76,99)(77,88)(78,96)(79,89)(85,136)(86,115)(91,94)(92,97)(95,107)(98,114)(102,141)(105,127)(106,118)(109,117)(111,119)(113,122)(126,131)(129,135)(130,140)(132,134)(133,137)(138,139);(14,17,22,21)(15,24,28,53)(16,30,34,54)(18,35,44,43)(19,36,49,48)(20,37,52,47)(23,55,59,58)(25,60,64,113)(26,61,66,105)(27,50,68,111)(29,69,72,71)(31,73,76,114)(32,74,78,107)(33,51,80,112)(38,81,90,103)(39,82,93,104)(40,83,96,95)(41,84,99,98)(42,85,102,97)(45,86,106,109)(46,87,108,110)(56,115,118,117)(57,116,119,67)(62,120,123,122)(63,88,126,137)(65,121,128,127)(70,129,130,79)(75,91,132,138)(77,131,133,125)(89,100,135,140)(92,101,136,141)(94,134,139,124)910 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6729);