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Cayley diagram

C7×SL(2,ℤ/4) #66648

order336
isoclinismisoclinic class of S4
automorphismAut(G) = C22×C3×S4 (order 288) · Inn(G) = S4 · Out(G) = C22×C3
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×SL(2,ℤ/4)
unambiguous semidirect productA4⋊(C4×C7)

Splittings

derivedA4 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7S4
commutator, hypocenterA4C4×C7
FrattiniC2C7×S4
Fitting, socleC23×C7D3
prime radicalC2×C7×A4C2
cosocleC2×A4C2×C7
LCSG ▷ A4
UCSC1 ◁ C2×C7

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10621477, (C7⋊S4)⋊C4, D7×SL(2,ℤ/4), …
maximal normal subgroup of#10621477, (C7⋊S4)⋊C4, D7×SL(2,ℤ/4), …
maximal quotient ofC7×A4⋊C8, C7×#859, C7×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 117);