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Cayley diagram

D49⋊C3 #64412

order294
isoclinismisoclinic class of D49⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD49⋊C3
unambiguous semidirect product(C49⋊C3)⋊C2

Splittings

Fitting, derived, SylowC49 ⋊ C2×C3
HallC49⋊C3 ⋊ C2
HallD49 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC49C2×C3
Frattini, socleC7AGL(1,𝔽7)
LCSG ▷ C49
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 49

A minimal faithful permutation representation — 2 generators on 49 points:

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

Polycyclic presentation:

2,3,7^2|c^7=d^6,c^a=c^6d,d^a=d^6,c^b=c^4d^5,d^b=d^4

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D49⋊C3, C3×D49⋊C3, C3⋊(D49⋊C3), …
maximal normal subgroup ofC2×D49⋊C3, C3×D49⋊C3, C5×D49⋊C3, …
maximal quotient of(C49⋊C3)⋊C4, C2×D49⋊C3, D49⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 1);