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Cayley diagram

C4⋊#672 #64682

order320
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C2×C16 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊#672

Splittings

SylowC5 ⋊ C4⋊C16

Subgroups

subgroupG/H
center, FrattiniC2×C8C2×D5
commutatorC2×C5C2×C16
FittingC4×C5×C8C2
socle, prime radicalC22×C5C2×C8
cosocleC2×C5×C8C22
hypercenterC4×C8D5
hypocenterC5C4⋊C16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C8 ◁ C4×C8

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

2^6,5|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bd,g^a=g^4

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10597930, #10597931, #10597958, …
maximal normal subgroup of#10597930, #10597931, #10597958, …
maximal quotient of#10597930, #10597931, C4⋊#3926, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 20);