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C7×#4005 #11764657

order1120
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#4005

Splittings

HallC5×C7 ⋊ #101
SylowC5 ⋊ C7×#101

Subgroups

subgroupG/H
centerC2×C7C22⋊AGL(1,𝔽5)
commutatorC22×C5C2×C4×C7
FrattiniC22C2×C7×AGL(1,𝔽5)
FittingC2×C4×C5×C7C4
socleC2×C5×C7C22⋊C4
prime radicalC22×C5×C7C2×C4
cosocleC2×Dic5C22×C7
hypercenterC2×C4×C7AGL(1,𝔽5)
hypocenterC5C7×#101
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 560

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3,4,5,6,7)(8,9,10,12,11)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(9,10,11,12)(13,14,16,19,17,20,25,28)(15,21,22,27,23,18,24,26)

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 263);