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C3×Dic49 #10596856

order588
isoclinismisoclinic class of D49
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic49

Splittings

derived, SylowC49 ⋊ C3×C4
HallC3×C49 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C3D49
commutator, hypocenterC49C3×C4
FrattiniC2×C7C3×D7
FittingC2×C3×C49C2
socle, prime radicalC2×C3×C7D7
cosocleC2×C49C2×C3
LCSG ▷ C49
UCSC1 ◁ C2×C3

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 588

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56);(4,6,5,7)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11924097, D147⋊C4, D3×Dic49, …
maximal normal subgroup of#11924097, D147⋊C4, D3×Dic49, …
maximal quotient ofC3×C49⋊C8, C2×C3×Dic49, C9×Dic49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(588, 4);