D₁₃×C₅₃ #13063392

order	1378
isoclinism	isoclinic class of D₁₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₃×C₅₃
unambiguous semidirect product	C₁₃⋊(C₂×C₅₃)

Splittings

Fitting, socle, Hall	C₁₃×C₅₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₁₃×C₅₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₅₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 689

A minimal faithful permutation representation — 2 generators on 66 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#261×C₅₃, #14583754, D₁₃×D₅₃, …
maximal normal subgroup of	#261×C₅₃, #14583754, D₁₃×D₅₃, …
maximal quotient of	Dic₁₃×C₅₃, C₂×D₁₃×C₅₃, C₃×D₁₃×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1378, 3);

D13×C53 #13063392