Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68000 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | #95⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4626 |
| commutator, hypocenter | #95⋊C3 | C4 |
| Frattini | C22×C4 | S4 |
| Fitting | #392 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical, cosocle | #392⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,8,5,2)(3,6,10,16)(4,20,9,11)(7,24,19,17)(12,18)(13,22,14,21)(15,23);(1,4,12)(2,7,11)(3,9,17)(5,14,23)(6,15,20)(8,19,22)(10,13,24)(16,18,21)Polycyclic presentation:
2^2,3,2^5|a^2=b,b^2=h,d^2=f,e^2=fg,c^a=c^2,d^a=efg,e^a=dh,f^a=fg,d^b=dfh,e^b=efgh,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#68000, #11711365, #11711370, … |
| maximal normal subgroup of | C2×#68000, #11711365, #11711370, … |
| maximal quotient of | #11710950, #11710961, #11710962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 576);