C₂²×#19097891 #34117463

order	20124
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂²×#19097891

Splittings

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

Subgroups

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

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 20124

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

Conjugacy classes & character table

2268 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 66);

C22×#19097891 #34117463