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C3×D4123 #37613101

order24738
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D4123

Splittings

Fitting, socle, HallC3×C7×C19×C31 ⋊ C2
derived, HallC7×C19×C31 ⋊ C2×C3
HallC3×C19×C31 ⋊ D7
HallC3×C7×C31 ⋊ D19
HallC19×C31 ⋊ C3×D7
HallC3×C7×C19 ⋊ D31
HallC7×C31 ⋊ C3×D19
HallC7×C19 ⋊ C3×D31
HallC3×C31 ⋊ D133
HallC3×C19 ⋊ D217
SylowC31 ⋊ C3×D133
HallC3×C7 ⋊ D589
SylowC19 ⋊ C3×D217
SylowC7 ⋊ C3×D589

Subgroups

subgroupG/H
center, hypercenterC3D4123
commutator, cosocle, hypocenterC7×C19×C31C2×C3
FrattiniC1G
Fitting, socleC3×C7×C19×C31C2
LCSG ▷ C7×C19×C31
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 12369

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

Conjugacy classes & character table

6189 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24738, 21);