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C11×ΓL(1,𝔽8)×D29 #27211535

order13398
automorphism?
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×ΓL(1,𝔽8)×D29
unambiguous semidirect productC29⋊(C2×C11×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D29)⋊(C3×C11)
unambiguous semidirect product(C7×C11×D29)⋊C3

Splittings

Fitting, socle, HallC7×C11×C29 ⋊ C2×C3
derived, HallC7×C29 ⋊ C2×C3×C11
HallC11×ΓL(1,𝔽8)×C29 ⋊ C2
HallC7×C11×D29 ⋊ C3
HallΓL(1,𝔽8)×C29 ⋊ C2×C11
HallC7×D29 ⋊ C3×C11
HallC11×C29 ⋊ C2×ΓL(1,𝔽8)
HallC7×C11 ⋊ C3×D29
SylowC29 ⋊ C2×C11×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C11×D29

Subgroups

subgroupG/H
center, hypercenterC11ΓL(1,𝔽8)×D29
commutator, cosocle, hypocenterC7×C29C2×C3×C11
FrattiniC1G
Fitting, socleC7×C11×C29C2×C3
LCSG ▷ C7×C29
UCSC1 ◁ C11

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 2233

A minimal faithful permutation representation — 2 generators on 47 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)(31,32,33)(34,36,35)(37,38,39,40,41,42,43,44,45,46,47);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,31,32,34,33,35,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC11×ΓL(1,𝔽8)⋊#1181, C11×AGL(1,𝔽7)×D29, C11⋊(ΓL(1,𝔽8)⋊#1181), …
maximal normal subgroup ofC11×ΓL(1,𝔽8)⋊#1181, C11×AGL(1,𝔽7)×D29, C11⋊(ΓL(1,𝔽8)⋊#1181), …
maximal quotient ofC11×ΓL(1,𝔽8)×Dic29, C2×C11×ΓL(1,𝔽8)×D29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13398, 26);