Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2262 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D16 |
| commutator, Frattini, cosocle | C16 | C23×C7 |
| socle | C2×C7 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ G | |
106 conjugacy classes of subgroups.
| 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)(10,21)(11,16)(12,30)(13,34)(14,20)(15,23)(17,32)(18,28)(19,29)(22,48)(24,54)(25,56)(26,49)(27,57)(31,45)(33,63)(35,42)(36,41)(37,50)(38,64)(39,53)(40,59)(43,69)(44,66)(46,61)(47,52)(51,68)(55,70)(58,67)(60,62)(65,71);(8,10)(9,15)(11,22)(12,31)(13,35)(14,26)(16,37)(17,42)(18,45)(19,47)(20,39)(21,30)(23,32)(24,28)(25,29)(27,58)(33,65)(34,38)(36,51)(40,54)(41,67)(43,70)(44,68)(46,71)(48,57)(49,63)(50,59)(52,69)(53,66)(55,61)(56,62)(60,64);(8,11,14,29)(9,16,20,19)(10,22,26,25)(12,27,33,60)(13,28,36,61)(15,37,39,47)(17,40,44,43)(18,41,46,34)(21,48,49,56)(23,50,53,52)(24,51,55,35)(30,57,63,62)(31,58,65,64)(32,59,66,69)(38,45,67,71)(42,54,68,70)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5497);