C₂×C₁₃×C₄₉⋊C₃ #16838422

order	3822
isoclinism	isoclinic class of C₄₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₁₃×C₄₉⋊C₃
unambiguous semidirect product	(C₂×C₄₉)⋊(C₃×C₁₃)

Splittings

Fitting, Hall	C₂×C₁₃×C₄₉ ⋊ C₃
derived, Sylow	C₄₉ ⋊ C₂×C₃×C₁₃
Hall	C₁₃×C₄₉ ⋊ C₂×C₃
Hall	C₂×C₄₉ ⋊ C₃×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₃	C₄₉⋊C₃
commutator, cosocle, hypocenter	C₄₉	C₂×C₃×C₁₃
Frattini	C₇	C₂×C₁₃×ΓL(1,𝔽₈)
Fitting	C₂×C₁₃×C₄₉	C₃
socle	C₂×C₇×C₁₃	ΓL(1,𝔽₈)
LCS	G ▷ C₄₉
UCS	C₁ ◁ C₂×C₁₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1274

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15)(17,19,23)(18,22,31)(20,26,38)(21,29,44)(24,35,33)(25,36,47)(27,40,58)(28,34,30)(32,49,48)(37,55,60)(39,57,50)(41,46,59)(42,52,53)(43,54,56)(45,61,51)(62,63,64);(16,46,29,57,54,39,62,18,47,20,64,44,33,40,22,38,28,24,41,51,35,32,56,43,37,26,53,52,31,17,59,45,23,63,55,48,21,19,27,34,60,50,49,30,25,42,36,58,61)

Conjugacy classes & character table

494 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₃⋊C₃×C₄₉⋊C₃, D₃×C₁₃×C₄₉⋊C₃, C₂×C₃×C₁₃×C₄₉⋊C₃, …
maximal normal subgroup of	C₂×C₁₃⋊C₃×C₄₉⋊C₃, C₂×C₃×C₁₃×C₄₉⋊C₃, C₂×C₅×C₁₃×C₄₉⋊C₃
maximal quotient of	C₂×C₁₃×C₄₉⋊C₉, D₃×C₁₃×C₄₉⋊C₃, C₂×C₃×C₁₃×C₄₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3822, 15);

C2×C13×C49⋊C3 #16838422