Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C26 |
| direct product | C13×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C2×C13×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C4×C13 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,26)(17,31,34,22)(18,23)(19,38)(20,25)(21,29)(24,41)(27,54,57,52)(28,53)(30,50)(32,61,65,45)(33,46,68,62)(35,69)(36,51)(37,49)(39,44)(40,48)(42,56,60,66)(43,59)(47,58)(55,71,75,74)(63,70,76,77)(64,72,67,73);(14,16,20,30)(15,21,25,44)(17,27,34,57)(18,35,37,59)(19,39,41,29)(22,42,31,60)(23,47,49,53)(24,50,38,26)(28,40,58,36)(32,63,65,76)(33,66,68,56)(43,51,69,48)(45,71,61,74)(46,54,62,52)(55,67,75,64)(70,73,77,72);(14,17)(15,22)(16,27)(18,32)(19,33)(20,34)(21,42)(23,45)(24,46)(25,31)(26,52)(28,55)(29,56)(30,57)(35,63)(36,64)(37,65)(38,62)(39,66)(40,67)(41,68)(43,70)(44,60)(47,71)(48,72)(49,61)(50,54)(51,73)(53,74)(58,75)(59,76)(69,77);(14,18,20,37)(15,23,25,49)(16,28,30,58)(17,32,34,65)(19,40,41,36)(21,43,44,69)(22,45,31,61)(24,51,38,48)(26,53,50,47)(27,55,57,75)(29,59,39,35)(33,67,68,64)(42,70,60,77)(46,73,62,72)(52,74,54,71)(56,76,66,63)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 15445);