Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = (#1020⋊C3)⋊C2 (order 576) · Inn(G) = C22⋊A4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽4) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽4) | C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C22 | |
80 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)(7,11,15)(8,16,13)(9,10,14);(2,9)(3,16)(6,15)(7,8)(10,11)(13,14);(1,6)(2,3)(4,14)(5,15)(7,11)(8,16)(9,10)(12,13)Polycyclic presentation:
3,2^6|b^a=c,c^a=bc,d^a=e,e^a=de,d^b=dg,e^b=efg,d^c=df,e^c=eg17 conjugacy classes — view the full character table.
| derived subgroup of | #85556, #85557, #85560, … |
| maximal subgroup of | #2202⋊C3, #85556, #85557, … |
| maximal normal subgroup of | #2202⋊C3, #85556, #85557, … |
| maximal quotient of | #73291, #73293, C2×#5465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1024);