Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #585 | D3 |
| socle | C2 | C2×GL(2,ℤ/4) |
| hypercenter | #41 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
297 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,34,42,3,17,45)(2,21,32,7,26,35)(4,31,38,16,22,27)(5,47,15,14,41,11)(6,30,39,9,40,46)(8,29,43,12,20,44)(10,19,25,24,33,18)(13,37,48,23,28,36);(1,2,8,13)(3,7,12,23)(4,9,15,25)(5,10,22,30)(6,11,18,16)(14,24,31,40)(17,26,20,28)(19,27,39,47)(21,29,37,34)(32,43,48,42)(33,38,46,41)(35,44,36,45);(3,11)(4,15)(5,19)(6,23)(7,18)(9,25)(10,27)(12,16)(14,32)(17,35)(20,36)(21,41)(22,39)(24,43)(26,44)(28,45)(29,33)(30,47)(31,48)(34,46)(37,38)(40,42)Polycyclic presentation:
2^4,3,2^3|b^2=h,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,d^c=dh,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11712836, #11712837, #11713094, … |
| maximal normal subgroup of | #11712836, #11712837, #11713094, … |
| maximal quotient of | #11711978, #11711979, #11711980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18120);