D₅₃×C₁₃⋊C₉ #27137151

order	12402
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅₃×C₁₃⋊C₉
unambiguous semidirect product	C₅₃⋊(C₂×C₁₃⋊C₉)
unambiguous semidirect product	(C₁₃×D₅₃)⋊C₉

Splittings

derived, Hall	C₁₃×C₅₃ ⋊ C₂×C₉
Hall	C₅₃×C₁₃⋊C₉ ⋊ C₂
Hall	C₁₃×D₅₃ ⋊ C₉
Sylow	C₅₃ ⋊ C₂×C₁₃⋊C₉
Sylow	C₁₃ ⋊ C₉×D₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₁₃⋊C₃×D₅₃
commutator, hypocenter	C₁₃×C₅₃	C₂×C₉
Fitting, socle, cosocle	C₃×C₁₃×C₅₃	C₂×C₃
prime radical	C₃×C₁₃×D₅₃	C₃
LCS	G ▷ C₁₃×C₅₃
UCS	C₁ ◁ C₃

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#14583755⋊C₉, D₅₃×D₁₃⋊C₉, C₁₃⋊C₉×#6157, …
maximal normal subgroup of	#14583755⋊C₉, D₅₃×D₁₃⋊C₉, C₁₃⋊C₉×#6157, …
maximal quotient of	C₁₃⋊C₉×Dic₅₃, C₂×D₅₃×C₁₃⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 14);

D53×C13⋊C9 #27137151