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Cayley diagram

#131⋊C3 #867

order96
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = A4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#131⋊C3

Splittings

Fitting, Sylow#131 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC8A4
commutator, hypocenterQ8C3×C4
FrattiniC4C2×A4
Fitting#131C3
socleC2C4×A4
prime radical#41⋊C3C2
cosocle#41C2×C3
LCSG ▷ Q8
UCSC1 ◁ C8

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2,3,2^4|a^2=c,c^2=f,d^2=f,e^2=f,d^b=e,e^b=de,e^d=ef

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of#73040, #73041, #73042, …
maximal subgroup of#4628, #504⋊C3, #5403, …
maximal normal subgroup of#4628, #504⋊C3, #5403, …
maximal quotient ofC8×SL(2,𝔽3), C2×#131⋊C3, #131⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 74);