Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C23×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #2614 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C23×C3×Q8 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #2614 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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)(15,23)(18,25)(21,27)(24,26);(2,3)(5,7)(9,11)(12,15)(13,18)(14,21)(16,23)(17,24)(19,25)(20,26)(22,27);(8,9)(10,11)(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(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=e,b^a=be,c^a=cf,c^b=cg,d^c=de,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10768354, #10768358, #10768376, … |
| maximal normal subgroup of | #10768354, #10768358, #10768376, … |
| maximal quotient of | #10721548, #10721604, #10721955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16645);