Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #532 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C4⋊Q8 | C2 |
| hypercenter | C4⋊Q8 | D3 |
| hypocenter | C3 | #532 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
310 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7)(8,10,11,14)(9,12,13,15);(8,9,11,13)(10,15,14,12);(4,5,6,7);(2,3)(10,14)(12,15)Polycyclic presentation:
2^6,3|b^2=f,c^2=e,d^2=e,d^a=de,g^a=g^2,d^b=df,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | #68417, #68421, #68426, … |
| maximal normal subgroup of | #68417, #68421, #68426, … |
| maximal quotient of | #74318, #74319, #74333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1284);