Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1804 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
376 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)(10,21,25,40)(11,26)(12,17)(13,33)(14,20)(15,23,39,56)(16,28)(18,31)(19,32)(22,50,53,71)(24,38,34,45)(27,59)(29,60)(30,46)(35,41)(36,44)(37,66,69,57)(42,64)(43,63)(47,62)(48,61,67,52)(49,65,70,55)(51,68,54,58);(8,10,14,25)(9,15)(11,22,29,53)(12,23,32,56)(13,34,36,24)(16,37)(17,21)(18,45)(19,40)(20,39)(26,58)(27,51,30,54)(28,61,63,52)(31,65,62,55)(33,49)(35,66,64,57)(38,47)(41,67)(42,48)(43,69)(44,70)(46,71)(50,59)(60,68);(8,11,14,29)(9,16,20,43)(10,22,25,53)(12,30,32,27)(13,35,36,64)(15,37,39,69)(17,42,19,41)(18,46,47,59)(21,48,40,67)(23,54,56,51)(24,57,34,66)(26,44,60,33)(28,62,63,31)(38,50,45,71)(49,58,70,68)(52,65,61,55)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5815);