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C52×ΓL(1,𝔽8)⋊C4 #14523564

order2100
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C52 · Schur multiplier (H2) = C5

Aliases

direct productC52×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C52)
unambiguous semidirect productDic7⋊(C3×C52)
unambiguous semidirect product(C5×Dic7)⋊(C3×C5)

Splittings

derived, SylowC7 ⋊ C3×C4×C52
HallC52×Dic7 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C4
HallC52×C7 ⋊ C3×C4
HallDic7 ⋊ C3×C52
HallΓL(1,𝔽8) ⋊ C4×C52

Subgroups

subgroupG/H
center, hypercenterC2×C52AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C52
FrattiniC2C52×AGL(1,𝔽7)
Fitting, socleC2×C52×C7C2×C3
prime radicalC2×C52×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C52
LCSG ▷ C7
UCSC1 ◁ C2×C52

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 700

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,4,5)(12,16,15,13,14,17)(18,19,20,21);(6,7,8,9,10)(11,12,15,16,14,17,13)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC52×(C49⋊C3)⋊C4, C52×C7×ΓL(1,𝔽8)⋊C4, C52×#10596874, …
maximal normal subgroup ofC52×C7×ΓL(1,𝔽8)⋊C4, C52×C11×ΓL(1,𝔽8)⋊C4, C52×C13×ΓL(1,𝔽8)⋊C4, …
maximal quotient ofC52×(C49⋊C3)⋊C4, C52×C7×ΓL(1,𝔽8)⋊C4, C52×#10596874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 44);