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C11×C22⋊(C52⋊C3) #15693221

order3300
isoclinismisoclinic class of C22⋊(C52⋊C3)
automorphism?
homologyAbelianization (H1) = C3×C11 · Schur multiplier (H2) = C2×C5

Aliases

direct productC11×C22⋊(C52⋊C3)
unambiguous semidirect productC22⋊(C11×C52⋊C3)
unambiguous semidirect productC52⋊(C11×A4)
unambiguous semidirect product(C22×C11)⋊(C52⋊C3)
unambiguous semidirect product(C52×C11)⋊A4

Splittings

Fitting, socle, HallC22×C52×C11 ⋊ C3
derived, HallC22×C52 ⋊ C3×C11
HallC52×C11 ⋊ A4
HallC22×C11 ⋊ C52⋊C3
SylowC52 ⋊ C11×A4
SylowC22 ⋊ C11×C52⋊C3

Subgroups

subgroupG/H
center, hypercenterC11C22⋊(C52⋊C3)
commutator, cosocle, hypocenterC22×C52C3×C11
FrattiniC1G
Fitting, socleC22×C52×C11C3
LCSG ▷ C22×C52
UCSC1 ◁ C11

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 330

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×C11×C22⋊(C52⋊C3), C11×C22⋊(C7⋊(C52⋊C3)), C121×C22⋊(C52⋊C3), …
maximal normal subgroup ofC7×C11×C22⋊(C52⋊C3), C121×C22⋊(C52⋊C3), C112×C22⋊(C52⋊C3)
maximal quotient ofC7×C11×C22⋊(C52⋊C3), C11×C22⋊(C7⋊(C52⋊C3)), C121×C22⋊(C52⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3300, 141);