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C5×#6398 #11764495

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

Aliases

direct productC5×#6398

Splittings

HallC5×C7 ⋊ #101
SylowC7 ⋊ C5×#101

Subgroups

subgroupG/H
centerC2×C5(C2×D7)⋊C4
commutatorC22×C7C2×C4×C5
FrattiniC2×C4C2×C5×D7
FittingC5×C7×M16C2
socleC2×C5×C7C22⋊C4
prime radicalC22×C5×C7C2×C4
cosocleC2×C4×C7C22×C5
hypercenterC5×M16D7
hypocenterC7C5×#101
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×M16

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

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, 101);