Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×D42 (order 384) · Inn(G) = C2×#67 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#67 |
| commutator, cosocle | C3×C4 | C23 |
| Frattini | C4 | C22×D3 |
| Fitting, prime radical | C3×#41 | C2 |
| socle | C2×C3 | C2×D4 |
| hypercenter | #41 | D3 |
| hypocenter | C3 | #137 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,10)(6,13)(7,15)(9,16)(11,12)(17,18);(4,6)(5,9)(7,14)(8,13)(10,18)(11,16)(12,17)(15,19);(2,3)(4,5,8,12)(6,9,13,17)(7,10,15,11)(14,18,19,16)Polycyclic presentation:
2^5,3|a^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^2,c^b=ce,d^c=de21 conjugacy classes — view the full character table.
| maximal subgroup of | #4825, #4826, #5141, … |
| maximal normal subgroup of | #4825, #4826, #5141, … |
| maximal quotient of | #4995, #4996, #5004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 158);