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S4×C49 #11924129

order1176
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C2×C49 · Schur multiplier (H2) = C2

Aliases

direct productS4×C49
unambiguous semidirect productA4⋊(C2×C49)

Splittings

FittingC22×C49 ⋊ D3
derivedA4 ⋊ C2×C49

Subgroups

subgroupG/H
center, hypercenterC49S4
commutator, hypocenterA4C2×C49
FrattiniC7C7×S4
FittingC22×C49D3
socleC22×C7D3×C7
prime radicalC7×S4C7
cosocleC7×A4C2×C7
LCSG ▷ A4
UCSC1 ◁ C49

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 196

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(1,2)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×7−1+2, S4×C49⋊C3, S4×D49
maximal normal subgroup ofS4×7−1+2, S4×C49⋊C3, S4×D49

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 44);