Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| prime radical | C2×#994 | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C2×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
313 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,33,20,18,21,34,10,16,22,7,8,32)(6,29,26,15,27,36,14,23,28,11,12,35)(9,17,19,31)(13,24,25,30);(5,6,10,14)(7,15,18,11)(8,12,21,27)(9,13,19,25)(16,29,33,23)(17,30,31,24)(20,26,22,28)(32,36,34,35);(1,2,3,4)(8,19)(9,21)(12,25)(13,27)(16,31)(17,33)(20,22)(23,30)(24,29)(26,28)(32,34)(35,36)Polycyclic presentation:
2^4,3,2^3|a^2=d,b^2=h,c^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712957, #11712960, #11712964, … |
| maximal normal subgroup of | #11712957, #11712960, #11712964, … |
| maximal quotient of | #11711788, #11711800, #11711809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18104);