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C52×#67 #10597149

order600
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC52×#67

Splittings

FittingC22×C3×C52 ⋊ C2
HallC3×C52 ⋊ D4
SylowC3 ⋊ C52×D4

Subgroups

subgroupG/H
centerC2×C52C2×D3
commutator, cosocleC2×C3C22×C52
FrattiniC2C2×C52×D3
FittingC22×C3×C52C2
socleC2×C3×C52C22
hypercenterC22×C52D3
hypocenterC3C52×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C52 ◁ C22×C52

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 300

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4,5)(11,12,13)(15,17);(6,7,8,9,10)(12,13)(14,15)(16,17)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#6780, C5×#6782, C5×D5×#67, …
maximal normal subgroup ofC5×#6780, C5×#6782, C5×D5×#67, …
maximal quotient ofC52×Dic3⋊C4, C52×(C2×D3)⋊C4, C52×#213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 130);