Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #577 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #577 | D3 |
| hypercenter | SD16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
256 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 256 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:
(1,19,16,7,17,24,6,9,18,2,4,23)(3,21,29,8,25,31,11,27,30,12,13,32)(5,10,15,20)(14,28,26,22);(2,8)(3,11)(7,12)(9,21)(10,22)(13,25)(14,26)(19,27)(20,28)(23,31)(24,32)(29,30);(2,7)(4,15)(5,17)(8,12)(9,10)(13,26)(14,25)(16,18)(19,20)(21,22)(27,28)(29,30)Polycyclic presentation:
2^4,3,2^3|c^2=h,d^2=h,f^2=h,g^2=h,c^a=ch,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=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85470, #11714981, #11714989, … |
| maximal normal subgroup of | C2×#85470, #11714981, #11714989, … |
| maximal quotient of | #11712268, #11712274, #11712277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18046);