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C7×(C2×AGL(1,𝔽5))⋊C4 #11764658

order1120
isoclinismisoclinic class of #700
automorphism?
homologyAbelianization (H1) = C42×C7 · Schur multiplier (H2) = C23

Aliases

direct productC7×(C2×AGL(1,𝔽5))⋊C4
unambiguous semidirect product(C2×AGL(1,𝔽5))⋊(C4×C7)

Splittings

FittingC2×C4×C5×C7 ⋊ C4
HallC5×C7 ⋊ #95
SylowC5 ⋊ C7×#95

Subgroups

subgroupG/H
centerC22×C7C2×AGL(1,𝔽5)
commutatorC2×C5C42×C7
FrattiniC22C2×C7×AGL(1,𝔽5)
FittingC2×C4×C5×C7C4
socleC22×C5×C7C2×C4
prime radicalC22×C7×D5C22
cosocleC22×D5C22×C7
hypercenterC2×C4×C7AGL(1,𝔽5)
hypocenterC5C7×#95
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C7 ◁ C2×C4×C7

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4,5,6,7)(8,9,10,12,11)(13,14,15,16)(18,20);(9,10,11,12)(17,18)(19,20)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×(C2×AGL(1,𝔽5))⋊C4, C7×C22⋊C4×AGL(1,𝔽5), #72218227, …
maximal normal subgroup ofΓL(1,𝔽8)×(C2×AGL(1,𝔽5))⋊C4, C7×C22⋊C4×AGL(1,𝔽5), #72218227, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 264);