Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #576 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #576 | D3 |
| hypercenter | Q16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,29,48,2,20,41,9,34,47,4,24,32)(3,30,44,18,38,39,8,28,40,14,25,45)(5,12,33,16,26,46,21,17,43,11,6,42)(7,31,23,13,15,22,19,37,35,10,27,36);(1,3,9,8)(2,10,4,13)(5,19,21,7)(6,15,26,27)(11,18,16,14)(12,28,17,30)(20,38,24,25)(22,41,36,32)(23,33,35,43)(29,37,34,31)(39,46,45,42)(40,48,44,47);(1,2)(3,13)(4,9)(5,11)(6,22)(7,18)(8,10)(12,23)(14,19)(15,32)(16,21)(17,35)(20,39)(24,45)(25,46)(26,36)(27,41)(28,43)(29,40)(30,33)(31,48)(34,44)(37,47)(38,42)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bh,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=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11714210, #11714211, #11714214, … |
| maximal normal subgroup of | #11714210, #11714211, #11714214, … |
| maximal quotient of | #11712364, #11712371, #11712383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18044);