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 | C24 | S4 |
| Fitting | #399 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical, cosocle | #399⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,14,4,5,10,11,12)(2,21,19,18,9,15,13,16)(6,7,23,22,17,20,24,8);(1,2,6)(3,8,23)(4,13,21)(5,9,17)(7,18,11)(10,22,24)(12,19,15)(14,20,16)Polycyclic presentation:
2^2,3,2^5|a^2=b,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×#68003, #11711364, #11711372, … |
| maximal normal subgroup of | C2×#68003, #11711364, #11711372, … |
| maximal quotient of | #11710950, #11710967, #11710968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 579);