Finite Group Databasebeta

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Cayley diagram

C2×#4339 #67034

order352
isoclinismisoclinic class of Dic22
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#4339

Splittings

FittingC22×C4×C11 ⋊ C2
SylowC11 ⋊ C2×#41

Subgroups

subgroupG/H
centerC2×C4C2×D11
commutator, cosocleC2×C11C24
FrattiniC2C23×D11
FittingC22×C4×C11C2
socleC22×C11C23
hypercenterC22×C4D11
hypocenterC11C2×#41
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C22×C4

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(4,8)(7,10)(11,12,14,16,18,20,21,19,17,15,13);(3,5,6,9)(4,7,8,10);(3,4)(5,7)(6,8)(9,10)(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#10622340, #10622346, #10622347, …
maximal normal subgroup of#10622340, #10622346, #10622347, …
maximal quotient ofC2×C4×Dic22, C2×#66923, #10623001, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 176);