C₉×C₁₃×D₅₃ #27137141

order	12402
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₉×C₁₃×D₅₃

Splittings

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

Subgroups

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

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

Conjugacy classes & character table

3276 conjugacy classes. The character table is too large to display.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 4);

C9×C13×D53 #27137141