C₅₃×D₁₃⋊C₉ #27137152

order	12402
isoclinism	isoclinic class of D₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₁₃⋊C₉
unambiguous semidirect product	D₁₃⋊(C₉×C₅₃)
unambiguous semidirect product	(C₁₃⋊C₉)⋊(C₂×C₅₃)
unambiguous semidirect product	(D₁₃×C₅₃)⋊C₉

Splittings

derived, Sylow	C₁₃ ⋊ C₂×C₉×C₅₃
Hall	C₅₃×C₁₃⋊C₉ ⋊ C₂
Hall	D₁₃×C₅₃ ⋊ C₉
Hall	C₁₃×C₅₃ ⋊ C₂×C₉
Hall	C₁₃⋊C₉ ⋊ C₂×C₅₃
Hall	D₁₃ ⋊ C₉×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅₃	D₁₃⋊C₃
commutator, hypocenter	C₁₃	C₂×C₉×C₅₃
Frattini	C₃	C₅₃×D₁₃⋊C₃
Fitting, socle	C₃×C₁₃×C₅₃	C₂×C₃
prime radical	C₃×D₁₃×C₅₃	C₃
cosocle	C₃×C₁₃	C₂×C₃×C₅₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₅₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6201

A minimal faithful permutation representation — 2 generators on 75 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(14,15,17,16,18,20,19,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(1,2,5,4,8,11,10,12,13,7,6,9,3)

Conjugacy classes & character table

1272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×#261⋊C₉, #14583754⋊C₉, D₅₃×D₁₃⋊C₉, …
maximal normal subgroup of	C₅₃×#261⋊C₉, #14583754⋊C₉, D₅₃×D₁₃⋊C₉, …
maximal quotient of	C₅₃×Dic₁₃⋊C₉, C₂×C₅₃×D₁₃⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 15);

C53×D13⋊C9 #27137152