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Cayley diagram

C22⋊D29 #6604

order232
isoclinismisoclinic class of Dic58
automorphismAut(G) = C22×AGL(1,𝔽29) (order 3248) · Inn(G) = C2×D29 · Out(G) = C4×C7
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊D29

Splittings

FittingC22×C29 ⋊ C2
SylowC29 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D29
commutator, socle, cosocleC2×C29C22
FittingC22×C29C2
hypercenterC22D29
hypocenterC29D4
LCSG ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 33 points:

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

Polycyclic presentation:

2^3,29|b^a=bc,d^a=d^28

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90805, D4×D29, #90807, …
maximal normal subgroup of#90805, D4×D29, #90807, …
maximal quotient of#90779, #90781, #90782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 8);