Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #578 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #578 | D3 |
| hypercenter | D8 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
257 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,17,18)(2,21,26,7,25,32)(3,13,30,12,27,29)(4,16,6)(8,9,31,11,19,24)(10,20)(14,28)(22,23);(1,2)(3,7)(4,9)(5,10)(6,11)(8,12)(13,19)(14,20)(15,23)(16,24)(17,25)(18,26)(21,27)(22,28)(29,31)(30,32);(4,15)(5,17)(9,23)(10,25)(13,28)(14,27)(16,18)(19,22)(20,21)(24,26)(29,30)(31,32)Polycyclic presentation:
2^4,3,2^3|d^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,d^b=dh,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11714210, #11714212, #11714215, … |
| maximal normal subgroup of | #11714210, #11714212, #11714215, … |
| maximal quotient of | #11712368, #11712374, #11712379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18049);