C₃²×D₁₃×C₅₃ #27137178

order	12402
isoclinism	isoclinic class of D₁₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₅₃ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×D₁₃×C₅₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₅₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₅₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃²×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 — 2 generators on 72 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)(70,71,72);(1,2,4,6,8,10,12,13,11,9,7,5,3)(67,68,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×#261×C₅₃, C₃²×#14583754, C₃²×D₁₃×D₅₃, …
maximal normal subgroup of	C₃²×#261×C₅₃, C₃²×#14583754, C₃²×D₁₃×D₅₃, …
maximal quotient of	C₃²×Dic₁₃×C₅₃, C₂×C₃²×D₁₃×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 41);

C32×D13×C53 #27137178