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C2×C5⋊D665 #27210922

order13300
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×C5⋊D665
unambiguous semidirect product(C2×C5)⋊D665
unambiguous semidirect product(C2×C7)⋊(C5⋊D95)
unambiguous semidirect product(C2×C19)⋊(C5⋊D35)

Splittings

Fitting, socleC2×C52×C7×C19 ⋊ C2
derived, HallC52×C7×C19 ⋊ C22
HallC52×C19 ⋊ C2×D7
HallC52×C7 ⋊ C2×D19
HallC7×C19 ⋊ C2×C5⋊D5
SylowC52 ⋊ C2×D133
SylowC19 ⋊ C2×C5⋊D35
SylowC7 ⋊ C2×C5⋊D95

Subgroups

subgroupG/H
center, hypercenterC2C5⋊D665
commutator, cosocle, hypocenterC52×C7×C19C22
FrattiniC1G
Fitting, socleC2×C52×C7×C19C2
LCSG ▷ C52×C7×C19
UCSC1 ◁ C2

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 6650

A minimal faithful permutation representation — 3 generators on 38 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22)(23,24)(25,26)(28,29)(30,31)(33,34)(35,36);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,23,25,26,24,22)(32,33,35,36,34)(37,38);(27,28,30,31,29)

Conjugacy classes & character table

3328 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 80);