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Cayley diagram

C2×C3×SL(2,ℤ/4) #64257

order288
isoclinismisoclinic class of S4
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = S4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×SL(2,ℤ/4)
unambiguous semidirect product(C2×A4)⋊(C3×C4)

Splittings

derivedA4 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC22×C3S4
commutator, hypocenterA4C2×C3×C4
FrattiniC2C2×C3×S4
Fitting, socleC24×C3D3
prime radicalC22×C3×A4C2
cosocleC2×A4C22×C3
LCSG ▷ A4
UCSC1 ◁ C22×C3

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×SL(2,ℤ/4), #10593238, SL(2,ℤ/4)⋊Dic3, …
maximal normal subgroup ofDic3×SL(2,ℤ/4), #10593238, SL(2,ℤ/4)⋊Dic3, …
maximal quotient ofC2×C3×A4⋊C8, C3×#5409, C3×C4×SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 905);