Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×D4 (order 192) · Inn(G) = D3×D4 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D4 |
| commutator, cosocle | C3×C4 | C23 |
| Frattini | C4 | C22×D3 |
| Fitting | C3×Q16 | C2 |
| socle | C2×C3 | C2×D4 |
| prime radical | #239 | C2 |
| hypercenter | Q16 | D3 |
| hypocenter | C3 | #137 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
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)(4,6,8,12)(5,9,11,18)(7,16,17,14)(10,19,15,13);(4,5,8,11)(6,13,12,19)(7,15,17,10)(9,16,18,14);(2,3)(6,12)(9,18)(13,19)(14,16)Polycyclic presentation:
2^5,3|b^2=e,c^2=e,d^2=e,c^a=ce,f^a=f^2,c^b=cd,d^b=de,d^c=de18 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#918, #5766, #5768, … |
| maximal normal subgroup of | C2×#918, #5766, #5768, … |
| maximal quotient of | #4789, #4791, #4794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 125);