C₁₃×C₂²⋊C₉×C₄₃ #34117462

order	20124
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₃×C₄₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₃×C₂²⋊C₉×C₄₃
unambiguous semidirect product	C₂²⋊(C₉×C₁₃×C₄₃)

Splittings

derived, Sylow	C₂² ⋊ C₉×C₁₃×C₄₃
Hall	C₂²×C₁₃×C₄₃ ⋊ C₉
Hall	C₂²×C₄₃ ⋊ C₉×C₁₃
Hall	C₂²×C₁₃ ⋊ C₉×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₃×C₄₃	A₄
commutator, hypocenter	C₂²	C₉×C₁₃×C₄₃
Frattini	C₃	A₄×C₁₃×C₄₃
Fitting, socle, prime radical	C₂²×C₃×C₁₃×C₄₃	C₃
cosocle	C₂²×C₃	C₃×C₁₃×C₄₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₃×C₄₃

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 10062

A minimal faithful permutation representation — 2 generators on 69 points:

(2,3,4)(5,6,8,7,9,11,10,12,13)(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)(57,58,59,60,61,62,63,64,65,66,67,68,69);(1,4)(2,3)

Conjugacy classes & character table

6708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 65);

C13×C22⋊C9×C43 #34117462