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Cayley diagram

C32 #94

order32
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C8 (order 16) · Inn(G) = C1 · Out(G) = C2×C8
holomorphG ⋊ Aut(G) = AGL(1,ℤ/32)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC32

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC16C2
socle, prime radicalC2C16
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 1 generator on 32 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

center ofM128, #2258, C3⋊C64, …
derived subgroup ofD64, SD128, Q128, …
Frattini subgroup ofC64, C2×C64, M128, …
Fitting subgroup ofC32×A5, #14514917
maximal subgroup ofC64, C2×C32, M64, …
maximal normal subgroup ofC64, C2×C32, M64, …
maximal quotient ofC64, C2×C32, C3⋊C32, …
outer automorphism group ofSD256, C3⋊(D97⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 1);