Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C22×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | 2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23 |
| Fitting | 2−1+4 | D3 |
| hypercenter | D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ D4 | |
140 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,4,15)(2,21,23,8,9,24)(3,25,27,11,12,28)(6,16,17)(7,18,30)(10,22)(13,26)(20,31,32);(1,3,6,11)(2,7,8,20)(4,12,16,25)(5,13,14,26)(9,18,21,31)(10,19,22,29)(15,27,17,28)(23,30,24,32);(1,2)(3,7)(4,10)(5,9)(6,8)(11,20)(12,19)(13,18)(14,21)(15,24)(16,22)(17,23)(25,29)(26,31)(27,32)(28,30)Polycyclic presentation:
2^3,3,2^3|b^2=g,e^2=g,f^2=g,c^a=cg,d^a=d^2,e^a=f,f^a=e,c^b=cg,e^d=fg,f^d=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #85471, #85472, #85474, … |
| maximal normal subgroup of | #85471, #85472, #85474, … |
| maximal quotient of | #85350, #85351, #85352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1486);