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Cayley diagram

C2×#733⋊C2 #66388

order324
isoclinismisoclinic class of #733⋊C2
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×#733⋊C2
unambiguous semidirect product#733⋊C22
unambiguous semidirect product(C2×#733)⋊C2

Splittings

FittingC2×#733 ⋊ C2
derived, Sylow#733 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#733⋊C2
commutator, cosocle, hypocenter#733C22
FrattiniC32C2×C3⋊D3
FittingC2×#733C2
socleC2×C3#272
LCSG ▷ #733
UCSC1 ◁ C2

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#733⋊C2, C4⋊(#733⋊C2), #10619685, …
maximal normal subgroup ofC4×#733⋊C2, C4⋊(#733⋊C2), #10619685, …
maximal quotient of#733⋊Q8, C4×#733⋊C2, C4⋊(#733⋊C2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 80);