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C2×C7×C3⋊D255 #36339591

order21420
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C7×C3⋊D255
unambiguous semidirect product(C2×C3)⋊(C7×D255)
unambiguous semidirect product(C2×C5)⋊(C7×C3⋊D51)
unambiguous semidirect product(C2×C17)⋊(C7×C3⋊D15)

Splittings

Fitting, socleC2×C32×C5×C7×C17 ⋊ C2
derived, HallC32×C5×C17 ⋊ C22×C7
HallC32×C5×C7×C17 ⋊ C22
HallC32×C7×C17 ⋊ C2×D5
HallC5×C7×C17 ⋊ C2×C3⋊D3
HallC32×C5×C7 ⋊ C2×D17
HallC32×C17 ⋊ C2×C7×D5
HallC7×C17 ⋊ C2×C3⋊D15
HallC5×C17 ⋊ C2×C7×C3⋊D3
HallC32×C7 ⋊ C2×D85
HallC32×C5 ⋊ C2×C7×D17
HallC5×C7 ⋊ C2×C3⋊D51
SylowC17 ⋊ C2×C7×C3⋊D15
SylowC32 ⋊ C2×C7×D85
SylowC5 ⋊ C2×C7×C3⋊D51

Subgroups

subgroupG/H
center, hypercenterC2×C7C3⋊D255
commutator, cosocle, hypocenterC32×C5×C17C22×C7
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C17C2
LCSG ▷ C32×C5×C17
UCSC1 ◁ C2×C7

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 10710

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

5376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 423);