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C7×#3993 #11764645

order1120
isoclinismisoclinic class of #3993
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#3993

Splittings

FittingC5×C7×C8 ⋊ C4
HallC5×C7 ⋊ #106
SylowC5 ⋊ C7×#106

Subgroups

subgroupG/H
centerC2×C7C4⋊AGL(1,𝔽5)
commutatorC4×C5C2×C4×C7
FrattiniC4C2×C7×AGL(1,𝔽5)
FittingC5×C7×C8C4
socleC2×C5×C7C4⋊C4
prime radicalC2×C7×D5D4
cosocleC4×D5C22×C7
hypercenterC7×C8AGL(1,𝔽5)
hypocenterC5C7×#106
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8

72 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)(13,14,15,17,16,18,19,20);(9,10,11,12)(14,17)(15,19)(18,20)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#3993, C7×#65731, #72218091, …
maximal normal subgroup ofΓL(1,𝔽8)×#3993, C7×#65731, #72218091, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 251);