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C5×C112×ΓL(1,𝔽8) #27163202

order12705
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C112 · Schur multiplier (H2) = C11

Aliases

direct productC5×C112×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C3×C5×C112)
unambiguous semidirect product(C5×C7)⋊(C3×C112)
unambiguous semidirect product(C5×C7×C11)⋊(C3×C11)

Splittings

Fitting, socle, HallC5×C7×C112 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C112
HallC7×C112 ⋊ C3×C5
HallC5×C7 ⋊ C3×C112

Subgroups

subgroupG/H
center, hypercenterC5×C112ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C3×C5×C112
FrattiniC1G
Fitting, socleC5×C7×C112C3
LCSG ▷ C7
UCSC1 ◁ C5×C112

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 4235

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3,4)(5,7,6)(19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34);(1,2,3,5,4,6,7)(8,9,10,11,12,13,14,15,16,17,18)

Conjugacy classes & character table

3025 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×C112×AGL(1,𝔽7), C5×C11×C11⋊AGL(1,𝔽7), C5×C11⋊(C11⋊AGL(1,𝔽7)), …
maximal normal subgroup ofC5×C112×AGL(1,𝔽7), C5×C11×C11⋊AGL(1,𝔽7), C5×C11⋊(C11⋊AGL(1,𝔽7)), …
maximal quotient ofC2×C5×C112×ΓL(1,𝔽8), C5×C112×C7⋊C9, C3×C5×C112×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12705, 17);