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Cayley diagram

C2×D5⋊AGL(1,𝔽5) #87916

order400
isoclinismisoclinic class of D5⋊AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×D5⋊AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C2×AGL(1,𝔽5))
unambiguous semidirect product#1043⋊C22

Splittings

Fitting, socleC2×C52 ⋊ C2×C4
derived, SylowC52 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2D5⋊AGL(1,𝔽5)
commutator, hypocenterC52C22×C4
FrattiniC1G
Fitting, socleC2×C52C2×C4
prime radicalC2×D52C2
cosocleC5⋊D5C23
LCSG ▷ C52
UCSC1 ◁ C2

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 3 generators on 12 points:

(3,4,7,5,6)(8,9)(11,12);(4,5,6,7)(9,10,11,12);(1,2)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊AGL(1,𝔽5), #11717304, #11717306, …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊AGL(1,𝔽5), #11717304, #11717306, …
maximal quotient of#11717353, #11717354, #11717355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 210);