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C7×#681 #10587882

order560
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#681

Splittings

HallC5×C7 ⋊ M16
SylowC5 ⋊ C7×M16

Subgroups

subgroupG/H
centerC4×C7C2×D5
commutatorC2×C5C2×C4×C7
FrattiniC4C2×C7×D5
FittingC2×C4×C5×C7C2
socleC2×C5×C7C2×C4
prime radicalC22×C5×C7C4
cosocleC4×C5C22×C7
hypercenterC2×C4×C7D5
hypocenterC5C7×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C4×C7 ◁ C2×C4×C7

40 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)(8,9,11,12,10)(14,17)(18,20);(9,10)(11,12)(13,14,15,18,16,17,19,20)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11764428, #11764430, #11764431, …
maximal normal subgroup of#11764428, #11764430, #11764431, …
maximal quotient ofC7×#3935, C7×C4⋊#168, C7×C22⋊#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(560, 54);