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C13×ΓL(1,𝔽8)×D37 #34118504

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

Aliases

direct productC13×ΓL(1,𝔽8)×D37

Splittings

Fitting, socle, HallC7×C13×C37 ⋊ C2×C3
derived, HallC7×C37 ⋊ C2×C3×C13
HallC13×ΓL(1,𝔽8)×C37 ⋊ C2
HallC7×C13×D37 ⋊ C3
HallΓL(1,𝔽8)×C37 ⋊ C2×C13
HallC7×D37 ⋊ C3×C13
HallC13×C37 ⋊ C2×ΓL(1,𝔽8)
HallC7×C13 ⋊ C3×D37
SylowC37 ⋊ C2×C13×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C13×D37

Subgroups

subgroupG/H
center, hypercenterC13ΓL(1,𝔽8)×D37
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:

(2,3)(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)(42,44,43)(45,46,47,48,49,50,51,52,53,54,55,56,57);(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)(38,39,40,42,41,43,44)

Conjugacy classes & character table

1300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20202, 34);