Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857⋊C2 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #392 | D3 |
| prime radical | #392⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,15,24,3,14,17,6,4,21,5,11,13)(2,18,12,7,19,20,9,8,23,10,16,22);(1,2,6,9)(3,10,5,7)(4,12,15,23)(8,17,18,13)(11,20,14,22)(16,21,19,24)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=f,e^2=fg,b^a=bh,c^a=c^2,d^a=ef,e^a=dfg,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 | #11711429, #11711441, C2×#72995, … |
| maximal normal subgroup of | #11711429, #11711441, C2×#72995, … |
| maximal quotient of | #11711090, #11711132, #11711240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5571);