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C5×#6410 #11764507

order1120
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#6410

Splittings

HallC5×C7 ⋊ #101
SylowC7 ⋊ C5×#101

Subgroups

subgroupG/H
centerC2×C5C22⋊Dic7
commutatorC22×C7C2×C4×C5
FrattiniC2×C4C2×C5×D7
FittingC2×C5×C7×Q8C2
socleC2×C5×C7C22⋊C4
prime radicalC22×C5×C7C2×C4
cosocleC2×C4×C7C22×C5
hypercenterC2×C5×Q8D7
hypocenterC7C5×#101
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×Q8

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

Conjugacy classes & character table

205 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 113);