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C13×(C37⋊C3)⋊D7 #34118552

order20202
automorphism?
homologyAbelianization (H1) = C2×C3×C13 · Schur multiplier (H2) = C1

Aliases

direct productC13×(C37⋊C3)⋊D7
unambiguous semidirect product(C37⋊C3)⋊(C13×D7)

Splittings

Fitting, socle, HallC7×C13×C37 ⋊ C2×C3
derived, HallC7×C37 ⋊ C2×C3×C13
HallC7×C13×C37⋊C3 ⋊ C2
HallC13×D259 ⋊ C3
HallC13×C37⋊C3 ⋊ D7
HallC7×C37⋊C3 ⋊ C2×C13
HallD259 ⋊ C3×C13
HallC13×C37 ⋊ C3×D7
HallC37⋊C3 ⋊ C13×D7
HallC7×C13 ⋊ D37⋊C3
SylowC37 ⋊ C3×C13×D7
SylowC7 ⋊ C13×D37⋊C3

Subgroups

subgroupG/H
center, hypercenterC13(C37⋊C3)⋊D7
commutator, cosocle, hypocenterC7×C37C2×C3×C13
FrattiniC1G
Fitting, socleC7×C13×C37C2×C3
LCSG ▷ C7×C37
UCSC1 ◁ C13

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 3367

A minimal faithful permutation representation — 2 generators on 57 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20202, 82);