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C2×C49⋊(D31⋊C5) #44477107

order30380
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C49⋊(D31⋊C5)
unambiguous semidirect product(C2×C49)⋊(D31⋊C5)
unambiguous semidirect product(C2×ΓL(1,𝔽32))⋊D49
unambiguous semidirect product(C2×D1519)⋊C5

Splittings

FittingC2×C31×C49 ⋊ C2×C5
derived, HallC31×C49 ⋊ C22×C5
HallC49×ΓL(1,𝔽32) ⋊ C22
HallC2×D1519 ⋊ C5
HallΓL(1,𝔽32) ⋊ C2×D49
SylowC49 ⋊ C2×D31⋊C5
SylowC31 ⋊ C2×C5×D49

Subgroups

subgroupG/H
center, hypercenterC2C49⋊(D31⋊C5)
commutator, cosocle, hypocenterC31×C49C22×C5
FrattiniC7C2×C7⋊(D31⋊C5)
FittingC2×C31×C49C2×C5
socleC2×C7×C31C5×D7
LCSG ▷ C31×C49
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 82
transitive degreeμt(G) = 3038

A minimal faithful permutation representation — 2 generators on 82 points:

(4,8,9,20,7,5,6,13,14,11)(10,25,26,27,12,16,17,29,31,19)(15,32,24,30,18,22,23,33,21,28)(35,37)(36,40)(38,43)(39,46)(41,49)(42,48)(44,53)(45,56)(47,58)(50,62)(51,61)(52,60)(54,55)(57,67)(59,69)(63,70)(64,72)(65,71)(66,75)(68,77)(73,78)(74,79)(76,81)(80,82);(1,2)(3,4,7,12,14,21,31,25,9,15,24,32,26,28,16,8,6,10,18,29,23,33,22,13,17,27,30,20,19,11,5)(34,48,72,82,74,52,37,40,61,78,76,65,43,46,49,70,68,66,53,56,58,62,59,57,55,54,67,69,50,47,45,44,75,77,63,41,39,38,71,81,73,51,36,35,60,79,80,64,42)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30380, 16);