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Cayley diagram

#548⋊C3 #5444

order192
isoclinismisoclinic class of #548⋊C3
automorphismAut(G) = C24×S4 (order 384) · Inn(G) = C22×A4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#548⋊C3

Splittings

Fitting, Sylow#548 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, cosocleC2×Q8C22×C3
Fitting#548C3
hypercenterC22⋊C4A4
hypocenterQ8C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C22⋊C4

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#85374, #85375, #85376, …
maximal normal subgroup of#85374, #85375, #85376, …
maximal quotient ofC22⋊C4×SL(2,𝔽3), #2341⋊C3, #2351⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1003);