Finite Group Databasebeta

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Cayley diagram

C2×(C2×D11)⋊C4 #66980

order352
isoclinismisoclinic class of Dic22
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

direct productC2×(C2×D11)⋊C4

Splittings

FittingC22×C4×C11 ⋊ C2
SylowC11 ⋊ C2×C22⋊C4

Subgroups

subgroupG/H
centerC23C2×D11
commutatorC2×C11C22×C4
FrattiniC22C22×D11
FittingC22×C4×C11C2
socleC23×C11C22
prime radicalC23×D11C2
cosocleC22×C11C23
hypercenterC22×C4D11
hypocenterC11C2×C22⋊C4
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C4

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 176

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10622106, #10622277, #10622278, …
maximal normal subgroup of#10622106, #10622277, #10622278, …
maximal quotient ofC2×#66869, #10622542, C4×(C2×D11)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 122);