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C5×C7⋊(C19⋊AGL(1,𝔽5)) #27210887

order13300
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C7⋊(C19⋊AGL(1,𝔽5))
unambiguous semidirect productC7⋊(C5×C19⋊AGL(1,𝔽5))
unambiguous semidirect productC19⋊(C5×C7⋊AGL(1,𝔽5))

Splittings

Fitting, socle, HallC52×C7×C19 ⋊ C4
derivedC5×C7×C19 ⋊ C4×C5
HallC52×C19 ⋊ Dic7
HallC52×C7 ⋊ Dic19
HallC7×C19 ⋊ C5×AGL(1,𝔽5)
SylowC52 ⋊ Dic133
SylowC19 ⋊ C5×C7⋊AGL(1,𝔽5)
SylowC7 ⋊ C5×C19⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC5C7⋊(C19⋊AGL(1,𝔽5))
commutator, hypocenterC5×C7×C19C4×C5
FrattiniC1G
Fitting, socleC52×C7×C19C4
prime radicalC5×C7×D5×C19C2
cosocleC7×D5×C19C2×C5
LCSG ▷ C5×C7×C19
UCSC1 ◁ C5

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 2660

A minimal faithful permutation representation — 2 generators on 36 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)(27,28,29,30,31)(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)(32,33,34,36,35)

Conjugacy classes & character table

1345 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 45);