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C7×C22⋊(C52⋊C3) #14523672

order2100
isoclinismisoclinic class of C22⋊(C52⋊C3)
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C2×C5

Aliases

direct productC7×C22⋊(C52⋊C3)
unambiguous semidirect productC22⋊(C7×C52⋊C3)
unambiguous semidirect productC52⋊(C7×A4)

Splittings

Fitting, socle, HallC22×C52×C7 ⋊ C3
derived, HallC22×C52 ⋊ C3×C7
HallC52×C7 ⋊ A4
HallC22×C7 ⋊ C52⋊C3
SylowC52 ⋊ C7×A4
SylowC22 ⋊ C7×C52⋊C3

Subgroups

subgroupG/H
center, hypercenterC7C22⋊(C52⋊C3)
commutator, cosocle, hypocenterC22×C52C3×C7
FrattiniC1G
Fitting, socleC22×C52×C7C3
LCSG ▷ C22×C52
UCSC1 ◁ C7

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 210

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3,4,5,6,7)(9,10,11)(12,13,15)(14,17,19)(16,21,24)(18,23,25)(20,26,22);(8,11)(9,10)(12,14,18,24,26)(13,17,23,16,22)(15,21,19,20,25)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC49×C22⋊(C52⋊C3), C7×C22⋊(C7⋊(C52⋊C3)), C72×C22⋊(C52⋊C3), …
maximal normal subgroup ofC49×C22⋊(C52⋊C3), C72×C22⋊(C52⋊C3), C7×C11×C22⋊(C52⋊C3), …
maximal quotient ofC49×C22⋊(C52⋊C3), C7×C22⋊(C7⋊(C52⋊C3)), C72×C22⋊(C52⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 152);