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 | |
254 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 254 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:
(1,23,16,10,17,24,6,8,18,2,4,22)(3,19,29,7,25,31,11,27,30,12,13,32)(5,9,15,21)(14,28,26,20);(2,7)(3,11)(8,19)(9,20)(10,12)(13,25)(14,26)(21,28)(22,31)(23,27)(24,32)(29,30);(4,15)(5,17)(8,21)(9,23)(13,26)(14,25)(16,18)(19,28)(20,27)(22,24)(29,30)(31,32)Polycyclic presentation:
2^4,3,2^3|c^2=h,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 | C2×#85469, #11714988, #11714992, … |
| maximal normal subgroup of | C2×#85469, #11714988, #11714992, … |
| maximal quotient of | #11712367, #11712376, #11712377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18045);