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C4×ΓL(1,𝔽8)×C25 #14523525

order2100
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C25 · Schur multiplier (H2) = C1

Aliases

direct productC4×ΓL(1,𝔽8)×C25
unambiguous semidirect product(C4×C7)⋊(C3×C25)
unambiguous semidirect product(C4×C7×C25)⋊C3

Splittings

Fitting, HallC4×C7×C25 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C25
HallC7×C25 ⋊ C3×C4
HallC4×C7 ⋊ C3×C25

Subgroups

subgroupG/H
center, hypercenterC4×C25ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C25
FrattiniC2×C5C2×C5×ΓL(1,𝔽8)
FittingC4×C7×C25C3
socle, cosocleC2×C5×C7C2×C3×C5
prime radicalC2×C5×ΓL(1,𝔽8)C2×C5
LCSG ▷ C7
UCSC1 ◁ C4×C25

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 700

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3,4)(6,7,8)(9,11,10)(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);(5,6,7,9,8,10,11)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC4×C25×C49⋊C3, C25×#10596873, C25×#10596872, …
maximal normal subgroup ofC4×C7×ΓL(1,𝔽8)×C25, C4×C11×ΓL(1,𝔽8)×C25, C4×C13×ΓL(1,𝔽8)×C25, …
maximal quotient ofC4×C25×C49⋊C3, C25×#10596873, C25×#10596872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 5);