C₂²⋊#19097891 #34117458

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

Aliases

unambiguous semidirect product

C₂²⋊#19097891

Splittings

derived, Hall	C₂²×C₁₃×C₄₃ ⋊ C₉
Hall	C₁₃×C₄₃ ⋊ C₂²⋊C₉
Hall	C₂²×C₄₃ ⋊ C₁₃⋊C₉
Hall	C₂²×C₁₃ ⋊ C₄₃⋊C₉
Sylow	C₄₃ ⋊ C₂²⋊(C₁₃⋊C₉)
Sylow	C₁₃ ⋊ C₂²⋊(C₄₃⋊C₉)
Sylow	C₂² ⋊ #19097891

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₂²⋊#13131976
commutator, hypocenter	C₂²×C₁₃×C₄₃	C₉
Fitting, socle, prime radical, cosocle	C₂²×C₃×C₁₃×C₄₃	C₃
LCS	G ▷ C₂²×C₁₃×C₄₃
UCS	C₁ ◁ C₃

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)(45,46,48)(47,50,53)(49,52,51)(54,56,55)(58,59,60)(61,62,64,63,65,67,66,68,69);(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)(44,45,47,46,49,53,50,54,56,48,51,55,52)(57,60)(58,59)

Polycyclic presentation:

3,2^2,13,43,3|a^3=f,b^a=c,c^a=bc,d^a=d^3,e^a=e^36

Conjugacy classes & character table

2244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 61);

C22⋊#19097891 #34117458