Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C4×D4 | C2 |
| hypercenter | C4×D4 | D3 |
| hypocenter | C3 | #547 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 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)(4,5,6,7);(9,13)(10,14);(8,10,11,14)(9,12,13,15);(2,3)(5,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^6,3|b^2=f,d^2=e,c^a=cf,d^a=de,g^a=g^2,c^b=cf45 conjugacy classes — view the full character table.
| maximal subgroup of | #68185, #68200, #68204, … |
| maximal normal subgroup of | #68185, #68200, #68204, … |
| maximal quotient of | #73456, #73464, #73467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1114);