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C7×#4147 #11765445

order1120
automorphism?
homologyAbelianization (H1) = C24×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×#4147

Splittings

FittingC2×C5×C7×Q8 ⋊ C2
HallC5×C7 ⋊ 2−1+4
SylowC5 ⋊ C7×2−1+4

Subgroups

subgroupG/H
centerC2×C7C23×D5
commutator, cosocleC2×C5C24×C7
FrattiniC2C23×C7×D5
FittingC2×C5×C7×Q8C2
socleC2×C5×C7C24
hypercenterC2×C7×Q8D5
hypocenterC5C7×2−1+4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C7 ◁ C2×C7×Q8

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 1051);