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, socle | C2 | C2×GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×2−1+4 | D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D4 | |
292 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 292 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,19,21)(2,23,28,9,20,31)(3,13,30,4,15,29)(5,18,7)(8,10,32,12,25,27)(11,26)(14,16)(17,24);(1,3,7,4)(2,8,12,9)(5,13,19,15)(6,14,22,16)(10,23,20,25)(11,24,17,26)(18,29,21,30)(27,31,28,32);(1,2)(3,8)(4,9)(5,17)(6,20)(7,12)(10,22)(11,19)(13,26)(14,25)(15,24)(16,23)(18,28)(21,27)(29,32)(30,31)Polycyclic presentation:
2^4,3,2^3|b^2=h,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713174, #11713175, #11713197, … |
| maximal normal subgroup of | #11713174, #11713175, #11713197, … |
| maximal quotient of | #11712062, #11712063, #11712064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18109);