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Cayley diagram

C22×D29 #6609

order232
isoclinismisoclinic class of D29
automorphismAut(G) = S4×AGL(1,𝔽29) (order 19488) · Inn(G) = D29 · Out(G) = C2×C7×S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D29
unambiguous semidirect productC29⋊C23
unambiguous semidirect product(C2×C29)⋊C22

Splittings

Fitting, socleC22×C29 ⋊ C2
derived, SylowC29 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D29
commutator, cosocle, hypocenterC29C23
FrattiniC1G
Fitting, socleC22×C29C2
LCSG ▷ C29
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 116

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2)(3,4)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33);(3,4)(5,6,8,10,12,14,16,18,20,22,24,26,28,30,32,33,31,29,27,25,23,21,19,17,15,13,11,9,7);(1,2)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90781, C22⋊#1181, C2×C4×D29, …
maximal normal subgroup of#90781, C22⋊#1181, C2×C4×D29, …
maximal quotient ofC2×Dic58, C2×C4×D29, C2×D116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 13);