Finite Group Databasebeta

← Prev (#65906) Next (#65908) →

Cayley diagram

C4×#713 #65907

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

Aliases

direct productC4×#713

Splittings

FittingC4×C5×Q8 ⋊ C2
SylowC5 ⋊ C4×#41

Subgroups

subgroupG/H
centerC2×C4C22×D5
commutatorC2×C5C23×C4
FrattiniC22C23×D5
FittingC4×C5×Q8C2
socle, cosocleC22×C5C24
prime radicalC2×#713C2
hypercenterC4×Q8D5
hypocenterC5C4×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×Q8

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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10599022, #10599023, #10599030, …
maximal normal subgroup of#10599022, #10599023, #10599030, …
maximal quotient ofC4×#4038, C4×#4039, C4×#4044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1245);