Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4656 |
| automorphism | Aut(G) = C29×D3 (order 3072) · Inn(G) = C22×D3 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C23 |
| Fitting | C2×C3×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D3 |
| hypocenter | C3 | #398 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
150 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,7)(8,10,11,9);(9,10)(12,14,15,19)(13,16,17,18);(2,3)(4,5)(6,7)(8,9,11,10)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=f,b^a=be,c^a=cd,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #73669, #73670, #73671, … |
| maximal normal subgroup of | #73669, #73670, #73671, … |
| maximal quotient of | #69410, #69411, #69414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 541);