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C7×D1443 #34118495

order20202
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D1443
unambiguous semidirect productC3⋊(C7×D481)
unambiguous semidirect productC13⋊(C7×D111)
unambiguous semidirect productC37⋊(C7×D39)

Splittings

Fitting, socle, HallC3×C7×C13×C37 ⋊ C2
derived, HallC3×C13×C37 ⋊ C2×C7
HallC7×C13×C37 ⋊ D3
HallC3×C7×C37 ⋊ D13
HallC13×C37 ⋊ D3×C7
HallC3×C7×C13 ⋊ D37
HallC7×C37 ⋊ D39
HallC3×C37 ⋊ C7×D13
HallC7×C13 ⋊ D111
HallC3×C13 ⋊ C7×D37
SylowC37 ⋊ C7×D39
HallC3×C7 ⋊ D481
SylowC13 ⋊ C7×D111
SylowC3 ⋊ C7×D481

Subgroups

subgroupG/H
center, hypercenterC7D1443
commutator, cosocle, hypocenterC3×C13×C37C2×C7
FrattiniC1G
Fitting, socleC3×C7×C13×C37C2
LCSG ▷ C3×C13×C37
UCSC1 ◁ C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 10101

A minimal faithful permutation representation — 2 generators on 60 points:

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

Conjugacy classes & character table

5061 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20202, 25);