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 | |
250 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 250 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:
(1,29,48,2,34,46,9,28,45,4,21,32)(3,27,40,20,31,42,8,35,47,13,24,39)(5,19,43,18,6,41,16,12,33,11,25,44)(7,30,23,10,15,22,14,38,36,17,26,37);(1,3,9,8)(2,10,4,17)(5,7,16,14)(6,15,25,26)(11,20,18,13)(12,27,19,35)(21,24,34,31)(22,32,37,46)(23,33,36,43)(28,38,29,30)(39,44,42,41)(40,45,47,48);(1,2)(3,10)(4,9)(5,11)(6,22)(7,20)(8,17)(12,23)(13,14)(15,32)(16,18)(19,36)(21,39)(24,44)(25,37)(26,46)(27,33)(28,40)(29,47)(30,48)(31,41)(34,42)(35,43)(38,45)Polycyclic presentation:
2^4,3,2^3|b^2=h,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×#85467, #11714981, #11714989, … |
| maximal normal subgroup of | C2×#85467, #11714981, #11714989, … |
| maximal quotient of | #11712270, #11712275, #11712279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18043);