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C3×C5×C7⋊C16 #13131985

order1680
isoclinismisoclinic class of D7
automorphism?
homologyAbelianization (H1) = C3×C5×C16 · Schur multiplier (H2) = C1

Aliases

direct productC3×C5×C7⋊C16

Splittings

derived, SylowC7 ⋊ C3×C5×C16
HallC3×C5×C7 ⋊ C16
HallC5×C7 ⋊ C3×C16
HallC3×C7 ⋊ C5×C16

Subgroups

subgroupG/H
center, hypercenterC3×C5×C8D7
commutator, hypocenterC7C3×C5×C16
FrattiniC8C3×C5×D7
FittingC3×C5×C7×C8C2
socle, prime radicalC2×C3×C5×C7C8
cosocleC7×C8C2×C3×C5
LCSG ▷ C7
UCSC1 ◁ C3×C5×C8

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1680

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 5);