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Cayley diagram

C2×C5×Q8 #718

order80
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C4×#1020⋊C2 (order 768) · Inn(G) = C22 · Out(G) = C2×C4×S4
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC2×C5×Q8

Subgroups

subgroupG/H
center, socle, prime radicalC22×C5C22
commutator, Frattini, cosocleC2C23×C5
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ G

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2)(3,4,5,6,7);(8,9,11,13)(10,14,15,12);(1,2)(8,10,11,15)(9,12,13,14)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10597951, #10597954, #10598033, …
Frattini subgroup of#13097683, #13097684, #13097748, …
Fitting subgroup of#3967, #3968, C2×#688, …
maximal subgroup of#3967, #3968, C5×#101, …
maximal normal subgroup of#3967, #3968, C5×#101, …
maximal quotient ofC2×C5×C4⋊C4, C4×C5×Q8, C5×C22⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 47);