C₅₃×C₃⋊D₃₉ #27137170

order	12402
isoclinism	isoclinic class of C₃⋊D₃₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅₃×C₃⋊D₃₉
unambiguous semidirect product	C₃⋊(C₅₃×D₃₉)
unambiguous semidirect product	C₁₃⋊(C₃⋊D₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃²×C₁₃×C₅₃ ⋊ C₂
derived, Hall	C₃²×C₁₃ ⋊ C₂×C₅₃
Hall	C₁₃×C₅₃ ⋊ C₃⋊D₃
Hall	C₃²×C₅₃ ⋊ D₁₃
Sylow	C₁₃ ⋊ C₃⋊D₃×C₅₃
Sylow	C₃² ⋊ D₁₃×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	C₃⋊D₃₉
commutator, cosocle, hypocenter	C₃²×C₁₃	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₅₃	C₂
LCS	G ▷ C₃²×C₁₃
UCS	C₁ ◁ 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:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,16)(18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72);(1,2,4,6,8,10,12,13,11,9,7,5,3)(17,18,19);(14,15,16)

Conjugacy classes & character table

3180 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×#90866, #37613711, D₃×C₅₃×D₃₉, …
maximal normal subgroup of	C₅₃×#90866, #37613711, D₃×C₅₃×D₃₉, …
maximal quotient of	C₅₃×C₃⋊Dic₃₉, C₂×C₅₃×C₃⋊D₃₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 33);

C53×C3⋊D39 #27137170