Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#131 ◁ G | |
206 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,15,21)(10,22,26,50)(11,27,30,47)(12,31,35,45)(13,20,14,19)(16,38,41,24)(17,36,43,29)(18,33,46,34)(23,51,55,68)(25,56,37,44)(28,59,62,42)(32,48,63,40)(39,60,67,71)(49,54,70,53)(52,57,65,66)(58,64,69,61);(8,10)(9,16)(11,23)(12,32)(13,25)(14,37)(15,26)(17,39)(18,44)(19,40)(20,48)(21,41)(22,31)(24,34)(27,57)(28,60)(29,53)(30,55)(33,38)(35,63)(36,54)(42,68)(43,67)(45,50)(46,56)(47,66)(49,58)(51,59)(52,61)(62,71)(64,65)(69,70);(8,11,15,30)(9,17,21,43)(10,23,26,55)(12,28,35,62)(13,36,14,29)(16,39,41,67)(18,42,46,59)(19,47,20,27)(22,49,50,70)(24,52,38,65)(25,54,37,53)(31,58,45,69)(32,60,63,71)(33,64,34,61)(40,66,48,57)(44,68,56,51)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5430);