Finite Group Databasebeta

← Prev (#11765119) Next (#11765121) →

C7×#4022 #11765120

order1120
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#4022

Splittings

FittingC42×C5×C7 ⋊ C2
HallC5×C7 ⋊ #126
SylowC5 ⋊ C7×#126

Subgroups

subgroupG/H
centerC22×C7C22×D5
commutator, cosocleC22×C5C23×C7
FrattiniC22C22×C7×D5
FittingC42×C5×C7C2
socleC22×C5×C7C23
prime radicalC22×C7×D5C22
hypercenterC42×C7D5
hypocenterC5C7×#126
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C7 ◁ C42×C7

120 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)(14,18)(17,20)(21,23,24,28)(22,26,25,27);(13,15,16,19)(14,17,18,20)(22,25)(26,27);(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,25)(26,28)

Conjugacy classes & character table

322 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 726);