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AGL(1,𝔽7)×#394 #72276672

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productAGL(1,𝔽7)×#394

Splittings

FittingC7×#394 ⋊ C2×C3
HallD7×#394 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#394
SylowC7 ⋊ C2×C3×#394

Subgroups

subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C24×C3
FittingC7×#394C2×C3
prime radicalC22×D4×AGL(1,𝔽7)C2
hypercenter#394AGL(1,𝔽7)
hypocenterC7C2×C3×#394
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

2504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#2679
maximal normal subgroup ofAGL(1,𝔽7)×#2679
maximal quotient ofAGL(1,𝔽7)×#1438

GAP

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

gap> G := Group((14,18,17,15,16,19), (2,4)(6,8)(13,14,17,18,16,19,15), (5,6,7,8)(9,10,11,12), (1,2)(3,4)(10,12));