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C3×D5425 #45654381

order32550
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D5425
unambiguous semidirect productC25⋊(C3×D217)

Splittings

Fitting, HallC3×C7×C25×C31 ⋊ C2
derived, HallC7×C25×C31 ⋊ C2×C3
HallC3×C25×C31 ⋊ D7
HallC25×C31 ⋊ C3×D7
HallC3×C7×C31 ⋊ D25
HallC3×C7×C25 ⋊ D31
HallC7×C31 ⋊ C3×D25
HallC7×C25 ⋊ C3×D31
HallC3×C31 ⋊ D175
HallC3×C25 ⋊ D217
SylowC31 ⋊ C3×D175
SylowC25 ⋊ C3×D217
HallC3×C7 ⋊ D775
SylowC7 ⋊ C3×D775

Subgroups

subgroupG/H
center, hypercenterC3D5425
commutator, cosocle, hypocenterC7×C25×C31C2×C3
FrattiniC5C3×D1085
FittingC3×C7×C25×C31C2
socleC3×C5×C7×C31D5
LCSG ▷ C7×C25×C31
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 16275

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

Conjugacy classes & character table

8142 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32550, 17);