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C2×(C61⋊C5)⋊D31 #49180688

order37820
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×(C61⋊C5)⋊D31
unambiguous semidirect product(C2×C61⋊C5)⋊D31

Splittings

Fitting, socleC2×C31×C61 ⋊ C2×C5
derived, HallC31×C61 ⋊ C22×C5
HallC31×C61⋊C5 ⋊ C22
HallC2×D1891 ⋊ C5
HallC61⋊C5 ⋊ C2×D31
SylowC61 ⋊ C2×C5×D31
SylowC31 ⋊ C2×D61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2(C61⋊C5)⋊D31
commutator, cosocle, hypocenterC31×C61C22×C5
FrattiniC1G
Fitting, socleC2×C31×C61C2×C5
LCSG ▷ C31×C61
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 94
transitive degreeμt(G) = 3782

A minimal faithful permutation representation — 2 generators on 94 points:

(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(35,39,40,44,61,36,37,45,47,55)(38,52,53,62,77,42,43,48,70,60)(41,64,65,66,78,49,50,69,82,83)(46,51,76,84,81,58,59,67,91,72)(54,68,89,73,74,57,71,85,56,90)(63,92,86,75,93,79,80,94,88,87);(1,2)(3,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5)(34,35,38,44,56,75,62,77,58,37,41,51,68,86,89,80,93,65,43,54,40,48,63,76,91,90,83,55,72,50,66,82,64,81,61,78,74,84,67,79,53,45,57,52,69,87,92,85,94,71,59,49,39,46,60,70,88,73,47,42,36)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37820, 68);