Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#1270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C22⋊C4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×D4×C53 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 79 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(3,4,5,7,6,8,9,10)(11,12)(13,17,18,15)(14,16)(19,20)(21,23)(22,25)(24,26);(11,13)(12,15)(14,18)(16,17)(20,23)(21,26)(24,25)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 546);