C₁₃×C₃⋊D₃×C₅₃ #27137172

order	12402
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₁₃×C₃⋊D₃×C₅₃
unambiguous semidirect product	C₃⋊(D₃×C₁₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₁₃×C₅₃ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₁₃×C₅₃
Hall	C₃²×C₅₃ ⋊ C₂×C₁₃
Hall	C₃²×C₁₃ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃×C₅₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₁₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₅₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₃×C₅₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 6201

A minimal faithful permutation representation — 3 generators on 72 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)(55,56)(58,59)(60,61,62,63,64,65,66,67,68,69,70,71,72);(57,58,59);(54,55,56)

Conjugacy classes & character table

4134 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃×#157×C₅₃, C₁₃×#14274232, C₅₃×#90865, …
maximal normal subgroup of	C₁₃×#157×C₅₃, C₁₃×#14274232, C₅₃×#90865, …
maximal quotient of	C₁₃×C₃⋊Dic₃×C₅₃, C₂×C₁₃×C₃⋊D₃×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 35);

C13×C3⋊D3×C53 #27137172