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Cayley diagram

C2×#6729 #91874

order480
isoclinismisoclinic class of #6728
automorphismAut(G) = D4×S5 (order 960) · Inn(G) = S5 · Out(G) = D4
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×#6729

Splittings

derivedSL(2,𝔽5) ⋊ C22

Subgroups

subgroupG/H
center, Fitting, socle, hypercenterC22S5
commutator, cosocle, hypocenterSL(2,𝔽5)C22
FrattiniC2C2×S5
LCSG ▷ SL(2,𝔽5)
UCSC1 ◁ C22

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 42 points:

(3,15,30,42,35,21)(4,7,24,37,13,5)(8,10,27,40,32,11)(9,12,17,36,25,19)(16,31,23,41,34,29)(18,38,28,22,26,20)(33,39);(1,2)(3,35,18,6,38,8,32,23,14,41)(4,22,26,25,27,12,29,16,37,30)(5,42,15,13,33,9,40,10,19,39)(7,36,20,21,34,17,24,31,11,28)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#6729, C4⋊#6729, #11751509, …
maximal normal subgroup ofC4×#6729, C4⋊#6729, #11751509, …
maximal quotient ofC4×#6729, C4⋊#6729, #11751505, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 950);