Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊C2 |
| commutator, prime radical, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #395 | D3 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,5,3,6,4,2)(7,8)(9,36,24,22,25,37,14,20,27,11,12,35)(10,33,31,19,26,40,18,29,32,15,16,39)(13,21,28,38)(17,30,23,34);(3,8)(5,6)(9,10,14,18)(11,19,22,15)(12,23,25,17)(13,26,28,16)(20,30,36,34)(21,29,38,33)(24,32,27,31)(35,39,37,40)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,c^a=c^2,d^a=eh,e^a=dh,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=deh,f^c=g,g^c=fg,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | #11711417, #11711428, C2×#73001, … |
| maximal normal subgroup of | #11711417, #11711428, C2×#73001, … |
| maximal quotient of | #11711130, #11711143, #11711242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5577);