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Cayley diagram

C7×C16 #1129

order112
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C4 (order 48) · Inn(G) = C1 · Out(G) = C22×C3×C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽7)×AGL(1,ℤ/16)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC112
abelianC7×C16
multiplicative(ℤ/113)ˣ, (ℤ/226)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC8C2×C7
socle, prime radicalC2×C7C8
LCSG ▷ C1
UCSC1 ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 1 generator on 23 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

center ofC7×M64, C7×#504, C7×C3⋊C32, …
derived subgroup ofD224, #89244, Dic112, …
Frattini subgroup ofC49⋊C32, C32×C49, Dic8624, …
Fitting subgroup ofC7⋊C32, D7×C16, #6372, …
maximal subgroup ofC7⋊C32, C7×C32, D7×C16, …
maximal normal subgroup ofC7⋊C32, C7×C32, D7×C16, …
maximal quotient ofC7×C32, C2×C7×C16, C7×C3⋊C16, …
automorphism group ofC113, C2×C113
outer automorphism group ofC113, C2×C113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 2);