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C2×C11×ΓL(1,𝔽8)×C29 #27211512

order13398
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C11×C29 · Schur multiplier (H2) = C1

Aliases

direct productC2×C11×ΓL(1,𝔽8)×C29
unambiguous semidirect product(C2×C7)⋊(C3×C11×C29)
unambiguous semidirect product(C2×C7×C11)⋊(C3×C29)
unambiguous semidirect product(C2×C7×C29)⋊(C3×C11)
unambiguous semidirect product(C2×C7×C11×C29)⋊C3

Splittings

Fitting, socle, HallC2×C7×C11×C29 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C11×C29
HallC7×C11×C29 ⋊ C2×C3
HallC2×C7×C29 ⋊ C3×C11
HallC7×C29 ⋊ C2×C3×C11
HallC2×C7×C11 ⋊ C3×C29
HallC7×C11 ⋊ C2×C3×C29
HallC2×C7 ⋊ C3×C11×C29

Subgroups

subgroupG/H
center, hypercenterC2×C11×C29ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C11×C29
FrattiniC1G
Fitting, socleC2×C7×C11×C29C3
LCSG ▷ C7
UCSC1 ◁ C2×C11×C29

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 4466

A minimal faithful permutation representation — 2 generators on 49 points:

(2,3,4)(5,7,6)(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);(1,2,3,5,4,6,7)

Conjugacy classes & character table

3190 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC4×C11×ΓL(1,𝔽8)×C29, C11×ΓL(1,𝔽8)×Dic29, ΓL(1,𝔽8)×C29×Dic11, …
maximal normal subgroup ofC4×C11×ΓL(1,𝔽8)×C29, C11×ΓL(1,𝔽8)×Dic29, ΓL(1,𝔽8)×C29×Dic11, …
maximal quotient ofC4×C11×ΓL(1,𝔽8)×C29, C22×C11×ΓL(1,𝔽8)×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 3);