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Cayley diagram

#41×Dic5 #66160

order320
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct product#41×Dic5

Splittings

SylowC5 ⋊ C4×#41

Subgroups

subgroupG/H
centerC2×C4C22×D5
commutatorC2×C5C23×C4
FrattiniC22C23×D5
Fitting, prime radicalC2×C5×#41C2
socle, cosocleC22×C5C24
hypercenterC2×#41D5
hypocenterC5C4×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×#41

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10600832, #10600924, Dic5×#104, …
maximal normal subgroup of#10600832, #10600924, Dic5×#104, …
maximal quotient ofDic5×#117, #10613958, #10613960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1498);