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Cayley diagram

C2×#1042 #6059

order200
isoclinismisoclinic class of #1042
automorphismAut(G) = C2×AGL(1,𝔽5)2 (order 800) · Inn(G) = #1042 · Out(G) = C2×C4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×#1042
unambiguous semidirect productD5⋊Dic5

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2#1042
commutator, hypocenterC52C2×C4
FrattiniC1G
Fitting, socleC2×C52C4
prime radicalC2×C5×D5C2
cosocleC5×D5C22
LCSG ▷ C52
UCSC1 ◁ C2

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic5×AGL(1,𝔽5), (C2×AGL(1,𝔽5))⋊D5, #87825, …
maximal normal subgroup ofDic5×AGL(1,𝔽5), (C2×AGL(1,𝔽5))⋊D5, #87825, …
maximal quotient ofD5⋊#168, #87849, C4×#1042, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 46);