C₉×D₁₃×C₅₃ #27137142

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

Aliases

direct product

C₉×D₁₃×C₅₃

Splittings

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

Subgroups

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

(2,3)(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);(1,2,4,6,8,10,12,13,11,9,7,5,3)

Conjugacy classes & character table

3816 conjugacy classes. The character table is too large to display.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 5);

C9×D13×C53 #27137142