C₁₃×C₂²⋊(C₄₃⋊C₉) #34117461

order	20124
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

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

Subgroups

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

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:

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

Conjugacy classes & character table

2340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 64);

C13×C22⋊(C43⋊C9) #34117461