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C3×D4085 #37587765

order24510
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D4085
unambiguous semidirect productC5⋊(C3×D817)

Splittings

Fitting, socle, HallC3×C5×C19×C43 ⋊ C2
derived, HallC5×C19×C43 ⋊ C2×C3
HallC3×C19×C43 ⋊ D5
HallC19×C43 ⋊ C3×D5
HallC3×C5×C43 ⋊ D19
HallC3×C5×C19 ⋊ D43
HallC5×C43 ⋊ C3×D19
HallC3×C43 ⋊ D95
HallC5×C19 ⋊ C3×D43
HallC3×C19 ⋊ D215
SylowC43 ⋊ C3×D95
SylowC19 ⋊ C3×D215
HallC3×C5 ⋊ D817
SylowC5 ⋊ C3×D817

Subgroups

subgroupG/H
center, hypercenterC3D4085
commutator, cosocle, hypocenterC5×C19×C43C2×C3
FrattiniC1G
Fitting, socleC3×C5×C19×C43C2
LCSG ▷ C5×C19×C43
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 12255

A minimal faithful permutation representation — 2 generators on 70 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(5,8)(6,7)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Conjugacy classes & character table

6132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24510, 12);