Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, prime radical, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #557 | D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×Q8 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
189 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 189 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,8)(2,5,6,7)(9,15,19,10,20,23,14,17,21,11,12,24)(13,18,22,16);(1,2,4,6)(3,7,8,5);(9,10,14,11)(12,18,20,16)(13,17,22,15)(19,24,21,23)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=d,c^2=dh,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11712310, #11712311, #11712322, … |
| maximal normal subgroup of | #11712310, #11712311, #11712322, … |
| maximal quotient of | #11712974, C4⋊C4×#226, #11713022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17990);