Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = D3×D12 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #137 |
129 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,4,5,3,6,11,14,12)(2,8,9,15,10,13,16,7)(17,18,19)(20,21,22);(1,3,6,12)(2,7,10,15)(4,14,11,5)(8,9,13,16)(18,19);(1,2)(3,7)(4,13)(5,9)(6,10)(8,11)(12,15)(14,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=d,d^2=e,c^a=ce,g^a=g^2,c^b=cd,d^b=de,f^b=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10594676, #10594677, C2×#63800, … |
| maximal normal subgroup of | #10594676, #10594677, C2×#63800, … |
| maximal quotient of | #10590670, #10590674, #10590680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 448);