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C3×Dic70 #11719804

order840
isoclinismisoclinic class of Dic70
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic70

Splittings

HallC3×C5×C7 ⋊ Q8
HallC5×C7 ⋊ C3×Q8
HallC3×C7 ⋊ Dic10
HallC3×C5 ⋊ Dic14
SylowC7 ⋊ C3×Dic10
SylowC5 ⋊ C3×Dic14

Subgroups

subgroupG/H
centerC2×C3C2×D35
commutator, cosocleC2×C5×C7C22×C3
FrattiniC2C2×C3×D35
FittingC3×C4×C5×C7C2
socle, prime radicalC2×C3×C5×C7C22
hypercenterC3×C4D35
hypocenterC5×C7C3×Q8
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2×C3 ◁ C3×C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 840

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3)(4,5,6,7)(8,11,10,9)(12,13,14,15,16)(17,18,19,20,21,22,23);(4,8,6,10)(5,9,7,11)(13,16)(14,15)(18,23)(19,22)(20,21)

Conjugacy classes & character table

219 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#10587846, C3×#10587848, C3×#10587851, …
maximal normal subgroup ofC3×#10587846, C3×#10587848, C3×#10587851, …
maximal quotient ofC3×#10587900, C3×C4⋊Dic35, C2×C3×Dic70

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 106);