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C5×#10615674 #72226557

order3200
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25

Aliases

direct productC5×#10615674

Splittings

FittingC2×C52×AGL(1,ℤ/8) ⋊ C2
SylowC52 ⋊ #3068

Subgroups

subgroupG/H
centerC2×C5C2×#4085
commutator, cosocleC2×C4×C5C24×C5
FrattiniC2×C4C23×C5×D5
FittingC2×C52×AGL(1,ℤ/8)C2
socleC2×C52C2×#121
hypercenterC2×C5×AGL(1,ℤ/8)D5
hypocenterC5C5×#3068
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×AGL(1,ℤ/8)

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2,3,4,5)(6,7,9,10,8)(12,17)(13,20)(14,23)(16,19)(21,22)(24,25);(12,17)(16,24)(18,26)(19,25);(11,13)(12,16)(14,22)(15,21)(17,24)(18,19)(20,23)(25,26);(7,8)(9,10)(11,12,14,18,15,17,23,26)(13,19,20,16,21,25,22,24)

Conjugacy classes & character table

365 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of#72226564
maximal normal subgroup of#72226564

GAP

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

gap> G := Group((1,2,3,4,5)(6,7,9,10,8)(12,17)(13,20)(14,23)(16,19)(21,22)(24,25), (12,17)(16,24)(18,26)(19,25), (11,13)(12,16)(14,22)(15,21)(17,24)(18,19)(20,23)(25,26), (7,8)(9,10)(11,12,14,18,15,17,23,26)(13,19,20,16,21,25,22,24));