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AΣL(1,𝔽16)×C71 #17985482

order4544
nilpotency class4
isoclinismisoclinic class of AΣL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C2×C4×C71 · Schur multiplier (H2) = C22

Aliases

direct productAΣL(1,𝔽16)×C71

Subgroups

subgroupG/H
center, socleC2×C71#99
commutatorC23C2×C4×C71
Frattini, cosocleC2×D4C22×C71
prime radical#120×C71C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C22×C71 ◁ C2×D4×C71 ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 568

A minimal faithful permutation representation — 2 generators on 79 points:

(1,2)(3,7,8,5)(4,6)(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);(2,5)(6,7)

Conjugacy classes & character table

923 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AΣL(1,𝔽16)×C71, C71×#2120, C71×#2121, …
maximal normal subgroup ofC2×AΣL(1,𝔽16)×C71, C71×#2120, C71×#2121, …
maximal quotient ofC71×#1316, C71×#1326, C71×#1343, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4544, 32);