Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C4×Q8 | C2 |
| prime radical | C22×D3 | C23 |
| hypercenter | C4×Q8 | D3 |
| hypocenter | C3 | #541 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,7,8,6)(5,9,10,11)(13,19)(17,25)(18,27)(24,26);(4,6,8,7)(5,9,10,11)(12,15,16,23)(13,18,19,27)(14,20,21,22)(17,24,25,26);(12,14,16,21)(13,17,19,25)(15,22,23,20)(18,26,27,24);(2,3)(4,5)(6,11)(7,9)(8,10)(12,13)(14,17)(15,18)(16,19)(20,24)(21,25)(22,26)(23,27)Polycyclic presentation:
2^6,3|b^2=e,c^2=ef,d^2=f,c^a=cf,d^a=de,g^a=g^2,c^b=ce42 conjugacy classes — view the full character table.
| maximal subgroup of | #79657, #79659, #79667, … |
| maximal normal subgroup of | #79657, #79659, #79667, … |
| maximal quotient of | #73625, #73626, #73628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1142);