Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #5464 |
| automorphism | Aut(G) order 36864 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽4) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽4) | C3 |
| Frattini, socle | C22 | C22⋊A4 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3,6)(4,5,13)(7,11,16)(8,12,14)(9,10,15);(2,9)(3,12)(6,16)(7,8)(10,11)(14,15);(1,6)(2,3)(4,15)(5,14)(7,12)(8,10)(9,11)(13,16)Polycyclic presentation:
3,2^6|b^a=c,c^a=bc,d^a=e,e^a=de,f^a=g,g^a=fg,d^b=df,e^b=efg,d^c=dfg,e^c=eg| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 12 | 12 | 12 | 12 | 12 | 64 | 64 |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 3 | 3 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
| χ4 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 |
| χ5 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 |
| χ6 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 |
| χ7 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 |
| χ8 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 |
| χ9 | 12 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| derived subgroup of | #85558, #85559, #11711670, … |
| maximal subgroup of | #1655⋊C3, #85558, #85559, … |
| maximal normal subgroup of | #1655⋊C3, #85558, #85559, … |
| maximal quotient of | #2679⋊C3, #2736⋊C3, #73290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1023);