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Cayley diagram

#41⋊C3 #231

order48
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×S4 (order 48) · Inn(G) = A4 · Out(G) = C22
holomorphG ⋊ Aut(G) = #72224794
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#41⋊C3

Splittings

Fitting, Sylow#41 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4A4
commutator, cosocle, hypocenterQ8C2×C3
Frattini, socleC2C2×A4
Fitting#41C3
LCSG ▷ Q8
UCSC1 ◁ C4

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2,3,2^3|a^2=e,c^2=e,d^2=e,c^b=d,d^b=cd,d^c=de

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

derived subgroup of#5403, #5406, #5407, …
maximal subgroup ofGU(2,𝔽3), #131⋊C3, #984, …
maximal normal subgroup ofGU(2,𝔽3), #131⋊C3, #984, …
maximal quotient ofC4×SL(2,𝔽3), C2×#41⋊C3, #41⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 33);