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Cayley diagram

C2×C113 #6573

order226
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C7×C16 (order 112) · Inn(G) = C1 · Out(G) = C7×C16
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽113)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC226
abelianC2×C113
multiplicative(ℤ/227)ˣ, (ℤ/454)ˣ

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 115
transitive degreeμt(G) = 226

A minimal faithful permutation representation — 1 generator on 115 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115)

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

center ofD4×C113, Q8×C113, Dic3×C113, …
derived subgroup ofDic226, D452, C22⋊D113, …
Frattini subgroup ofDic12769
Fitting subgroup ofDic113, C2×D113, #11744753, …
socle ofDic113, C4×C113, C2×D113, …
maximal subgroup ofDic113, C4×C113, C2×D113, …
maximal normal subgroup ofDic113, C4×C113, C2×D113, …
maximal quotient ofC4×C113, C22×C113, D3×C113, …
automorphism group ofC227, C2×C227
outer automorphism group ofC227, C2×C227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(226, 2);