Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2658 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#124 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | C2×#124 | D3 |
| hypocenter | C3 | #2658 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
612 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(5,7)(9,11)(15,23)(18,25)(21,27)(24,26);(8,9,10,11)(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(2,3)(4,5,6,7)(12,13,16,19)(14,20,22,17)(15,18,23,25)(21,26,27,24)Polycyclic presentation:
2^7,3|a^2=e,b^2=efg,d^2=ef,b^a=bef,c^a=cf,h^a=h^2,c^b=cg,d^c=def60 conjugacy classes — view the full character table.
| maximal subgroup of | #10696569, #10696573, #10696625, … |
| maximal normal subgroup of | #10696569, #10696573, #10696625, … |
| maximal quotient of | #10721714, #10721716, #10721720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15556);