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C3×D4255 #38784281

order25530
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D4255
unambiguous semidirect productC5⋊(C3×D851)
unambiguous semidirect productC23⋊(C3×D185)

Splittings

Fitting, socle, HallC3×C5×C23×C37 ⋊ C2
derived, HallC5×C23×C37 ⋊ C2×C3
HallC3×C23×C37 ⋊ D5
HallC23×C37 ⋊ C3×D5
HallC3×C5×C37 ⋊ D23
HallC3×C5×C23 ⋊ D37
HallC5×C37 ⋊ C3×D23
HallC5×C23 ⋊ C3×D37
HallC3×C37 ⋊ D115
HallC3×C23 ⋊ D185
SylowC37 ⋊ C3×D115
SylowC23 ⋊ C3×D185
HallC3×C5 ⋊ D851
SylowC5 ⋊ C3×D851

Subgroups

subgroupG/H
center, hypercenterC3D4255
commutator, cosocle, hypocenterC5×C23×C37C2×C3
FrattiniC1G
Fitting, socleC3×C5×C23×C37C2
LCSG ▷ C5×C23×C37
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 12765

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

Conjugacy classes & character table

6387 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25530, 9);