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C3×D4495 #41004192

order26970
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D4495
unambiguous semidirect productC5⋊(C3×D899)
unambiguous semidirect productC29⋊(C3×D155)

Splittings

Fitting, socle, HallC3×C5×C29×C31 ⋊ C2
derived, HallC5×C29×C31 ⋊ C2×C3
HallC3×C29×C31 ⋊ D5
HallC29×C31 ⋊ C3×D5
HallC3×C5×C31 ⋊ D29
HallC3×C5×C29 ⋊ D31
HallC5×C31 ⋊ C3×D29
HallC5×C29 ⋊ C3×D31
HallC3×C31 ⋊ D145
HallC3×C29 ⋊ D155
SylowC31 ⋊ C3×D145
SylowC29 ⋊ C3×D155
HallC3×C5 ⋊ D899
SylowC5 ⋊ C3×D899

Subgroups

subgroupG/H
center, hypercenterC3D4495
commutator, cosocle, hypocenterC5×C29×C31C2×C3
FrattiniC1G
Fitting, socleC3×C5×C29×C31C2
LCSG ▷ C5×C29×C31
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 13485

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

6747 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26970, 11);