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Cayley diagram

C4⋊(#41⋊C3) #5446

order192
isoclinismisoclinic class of #548⋊C3
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊(#41⋊C3)
unambiguous semidirect product#550⋊C3

Splittings

Fitting, Sylow#550 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, cosocleC2×Q8C22×C3
Fitting#550C3
hypercenterC4⋊C4A4
hypocenterQ8C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C4⋊C4

87 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:

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

Polycyclic presentation:

2^2,3,2^4|a^2=g,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#73089, #73090, #73091, …
maximal normal subgroup of#73089, #73090, #73091, …
maximal quotient of#2341⋊C3, C4⋊C4×SL(2,𝔽3), #3391⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1005);