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C3×#27131472 #49138795

order36414
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C17

Aliases

direct productC3×#27131472

Splittings

Fitting, socleC3×C7×C172 ⋊ C2×C3
derived, HallC7×C172 ⋊ C2×C32
HallC3×C7×C172⋊C3 ⋊ C2
HallC7⋊(C17⋊D17) ⋊ C32
HallC3×C172⋊C3 ⋊ D7
SylowC172 ⋊ C32×D7
SylowC7 ⋊ C3×(C17⋊D17)⋊C3

Subgroups

subgroupG/H
center, hypercenterC3#27131472
commutator, cosocle, hypocenterC7×C172C2×C32
FrattiniC1G
Fitting, socleC3×C7×C172C2×C3
LCSG ▷ C7×C172
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 1071

A minimal faithful permutation representation — 2 generators on 61 points:

(5,6)(7,8)(9,10)(11,12,14)(13,27,28,17,18,40)(15,37,39,22,23,32)(16,21,45,25,26,36)(19,29,20,30,31,33)(24,53,56,41,42,61)(34,59,44,52,50,38)(35,55,47,48,54,43)(46,60,49,51,58,57);(1,2,3)(4,5,7,9,10,8,6)(11,13,19,32,50,60,61,36,55,54,45,56,58,59,39,30,17)(12,15,24,43,57,31,27,38,16,25,44,18,29,49,47,41,22)(14,20,34,53,48,26,46,23,40,28,37,51,21,35,42,52,33)

Conjugacy classes & character table

1053 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 90);