Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#20598 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3×D4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
5982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,5)(6,7)(8,9,10,11)(12,13)(14,16)(15,17)(18,19);(13,16)(14,19)(17,18);(5,7);(13,17)(16,18)192 conjugacy classes — view the full character table.
| Fitting subgroup of | #72248362, #72248364, #72248396, … |
| maximal subgroup of | C3×#421068, #72248362, #72248364, … |
| maximal normal subgroup of | C3×#421068, #72248362, #72248364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13586);