Finite Group Databasebeta

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Cayley diagram

C2×#3996 #65723

order320
isoclinismisoclinic class of #3993
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#3996

Splittings

SylowC5 ⋊ C2×#108

Subgroups

subgroupG/H
centerC22C4⋊AGL(1,𝔽5)
commutatorC4×C5C22×C4
FrattiniC4C22×AGL(1,𝔽5)
FittingC2×C5×C8C4
socleC22×C5C4⋊C4
prime radicalC22×D5Q8
cosocleC4×D5C23
hypercenterC2×C8AGL(1,𝔽5)
hypocenterC5C2×#108
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598620, #10598631, #10598667, …
maximal normal subgroup of#10598620, #10598631, #10598667, …
maximal quotient of#10603815, #10603817, #10603818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1061);