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C2×#72215772 #72228710

order4116
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×#72215772

Splittings

FittingC2×7−1+2 ⋊ C2×C3
derivedC49 ⋊ C22×C3×C7
HallC2×D49⋊C7 ⋊ C3
Hall(C49⋊C3)⋊C7 ⋊ C22
Sylow7−1+2 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC2#72215772
commutator, cosocle, hypocenterC49C22×C3×C7
FrattiniC7C2×C7×AGL(1,𝔽7)
FittingC2×7−1+2C2×C3
socleC2×C7C7×AGL(1,𝔽7)
LCSG ▷ C49
UCSC1 ◁ C2

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 98

A minimal faithful permutation representation — 2 generators on 51 points:

(2,19,41,24,39,49,26,11,20,32,25,44,38,45,14,12,21,22,18,37,34,4,7,46,8,33,36,47,5,35,17,40,48,30,23,15,6,31,43,13,16,42)(3,27,28,9,10,29);(1,8,39,42,36,45,4,3,26,21,15,48,19,13,10,17,5,35,43,40,12,29,38,16,46,41,37,30,28,6,25,22,20,11,32,27,18,7,44,14,31,47,9,2,23,49,34,33,24)(50,51)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#72215772, C22×#72215772, C4×#72215772, …
maximal normal subgroup ofC22×#72215772, C4×#72215772, #72275522, …
maximal quotient ofD3×#72215772, C22×#72215772, C4×#72215772, …
automorphism group ofDic49, C2×D49
holomorph ofC2×C49

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,19,41,24,39,49,26,11,20,32,25,44,38,45,14,12,21,22,18,37,34,4,7,46,8,33,36,47,5,35,17,40,48,30,23,15,6,31,43,13,16,42)(3,27,28,9,10,29), (1,8,39,42,36,45,4,3,26,21,15,48,19,13,10,17,5,35,43,40,12,29,38,16,46,41,37,30,28,6,25,22,20,11,32,27,18,7,44,14,31,47,9,2,23,49,34,33,24)(50,51));