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Cayley diagram

#4133⋊C3 #91888

order480
isoclinismisoclinic class of #4133⋊C3
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = A4×AGL(1,𝔽5) · Out(G) = C22
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#4133⋊C3

Splittings

Hall#4133 ⋊ C3
SylowC5 ⋊ #131⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×AGL(1,𝔽5)
commutator, Fitting, hypocenterC5×Q8C3×C4
socleC2×C5C4×A4
prime radicalC5⋊(#41⋊C3)C2
cosocle#713C2×C3
LCSG ▷ C5×Q8
UCSC1 ◁ C2

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(2,3,4,5)(6,7,8,12,11,15,18,25)(9,26,17,23,21,29,31,36,10,13,30,24,20,28,33,35,19,14,16,34,22,27,32,37);(1,2,3,5,4)(6,21,11,19)(7,28,15,26)(8,32,18,30)(9,10,20,22)(12,36,25,34)(13,14,27,29)(16,17,31,33)(23,24,35,37)

Polycyclic presentation:

2,3,2^3,5,2|a^2=c,c^2=g,d^2=g,e^2=g,f^a=f^2,d^b=de,e^b=d,f^c=f^4,e^d=eg

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11756699, #11756700, #11756701, …
maximal normal subgroup of#11756699, #11756700, #11756701, …
maximal quotient ofSL(2,𝔽3)×#170, C2×#4133⋊C3, #4133⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 964);