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C52×C7×A4 #14523670

order2100
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C3×C52×C7 · Schur multiplier (H2) = C2×C5

Aliases

direct productC52×C7×A4
unambiguous semidirect productC22⋊(C3×C52×C7)
unambiguous semidirect product(C22×C5)⋊(C3×C5×C7)

Splittings

Fitting, socle, HallC22×C52×C7 ⋊ C3
derived, SylowC22 ⋊ C3×C52×C7
HallC22×C52 ⋊ C3×C7
HallC22×C7 ⋊ C3×C52

Subgroups

subgroupG/H
center, hypercenterC52×C7A4
commutator, cosocle, hypocenterC22C3×C52×C7
FrattiniC1G
Fitting, socleC22×C52×C7C3
LCSG ▷ C22
UCSC1 ◁ C52×C7

80 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)(11,12,13,14,15,16,17)(19,20,21);(6,7,8,9,10)(18,19,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC52×A4×C49, C52×C7×C22⋊ΓL(1,𝔽8), C52×C72×A4, …
maximal normal subgroup ofC52×A4×C49, C52×C72×A4, C52×C7×C11×A4, …
maximal quotient ofC52×A4×C49, C52×C7×C22⋊ΓL(1,𝔽8), C52×C72×A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 150);