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C3×Dic56 #10620558

order672
isoclinismisoclinic class of D112
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic56

Splittings

HallC3×C7 ⋊ Q32
SylowC7 ⋊ C3×Q32

Subgroups

subgroupG/H
centerC2×C3D56
commutator, cosocleC7×C8C22×C3
FrattiniC8C2×C3×D7
FittingC3×C7×C16C2
socle, prime radicalC2×C3×C7D8
hypercenterC3×C16D7
hypocenterC7C3×Q32
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 672

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

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

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050850, #13050852, #13050855, …
maximal normal subgroup of#13050850, #13050852, #13050855, …
maximal quotient ofC3×#89299, C3×#89298, C2×C3×Dic56

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 140);