Finite Group Databasebeta

← Prev (#5439) Next (#5441) →

Cayley diagram

#520⋊C3 #5440

order192
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#520⋊C3

Splittings

Fitting, Sylow#520 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, hypocenterQ8C2×C3×C4
Fitting#520C3
prime radicalC2×#41⋊C3C2
cosocleC2×Q8C22×C3
hypercenterC4⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C22 ◁ C4⋊C4

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4)(6,10)(7,8,13)(11,18,19,17,12,20)(14,16,15);(5,19)(6,15)(7,18)(8,11)(9,20)(10,13)(12,14)(16,17)

Polycyclic presentation:

2^2,3,2^4|a^2=g,b^2=d,e^2=g,f^2=g,b^a=bg,e^c=f,f^c=ef,f^e=fg

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73075, #73076, #73077, …
maximal normal subgroup of#73075, #73076, #73077, …
maximal quotient of#73167, #2341⋊C3, C4⋊C4×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 999);