Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | Dic3×C4⋊C4 |
| Sylow | C3 ⋊ C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator | C2×C3 | C2×C42 |
| Fitting | C2×C3×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C23 |
| hypercenter | C2×C4⋊C4 | D3 |
| hypocenter | C3 | C4×C4⋊C4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
194 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 194 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) = 192 |
A minimal faithful permutation representation — 3 generators on 15 points:
(9,10,11,12)(13,14,15);(5,6,7,8)(10,12);(1,2,3,4)(14,15)60 conjugacy classes — view the full character table.
| maximal subgroup of | #68780, #68781, #68782, … |
| maximal normal subgroup of | #68780, #68781, #68782, … |
| maximal quotient of | Dic3×#95, #69401, #69407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 533);