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C7×#4065 #11765163

order1120
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×#4065

Splittings

FittingC5×C7×Q16 ⋊ C2
HallC5×C7 ⋊ #135
SylowC5 ⋊ C7×#135

Subgroups

subgroupG/H
centerC2×C7D4×D5
commutator, cosocleC4×C5C23×C7
FrattiniC4C22×C7×D5
FittingC5×C7×Q16C2
socleC2×C5×C7C2×D4
hypercenterC7×Q16D5
hypocenterC5C7×#135
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×Q16

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 769);