Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C2×2−1+4 | D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
312 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,15,16)(6,11,20,10,17,19)(7,14,9)(12,18);(5,6)(7,11)(8,12)(9,10)(13,18)(14,19)(15,17)(16,20);(1,2)(3,4)(6,10)(7,13)(8,15)(11,12)(14,16)(17,18)Polycyclic presentation:
2^4,3,2^3|f^2=h,g^2=h,b^a=bh,c^a=cd,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85539, #11714692, #11714695, … |
| maximal normal subgroup of | C2×#85539, #11714692, #11714695, … |
| maximal quotient of | #11712034, #11712039, #11712044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18115);