C₂×#23758451 #34117442

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

Aliases

direct product

C₂×#23758451

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₂²×C₉
Hall	#19097891 ⋊ C₂²
Hall	C₂×C₁₃×D₄₃ ⋊ C₉
Hall	C₂×D₄₃ ⋊ C₁₃⋊C₉
Sylow	C₄₃ ⋊ C₂²×C₁₃⋊C₉
Sylow	C₁₃ ⋊ C₂×D₄₃⋊C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#16809809
commutator, hypocenter	C₁₃×C₄₃	C₂²×C₉
Frattini	C₃	C₂×#16809809
Fitting, socle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
prime radical	C₂×C₃×C₁₃×D₄₃	C₃
cosocle	C₃×C₁₃×C₄₃	C₂²×C₃
LCS	G ▷ C₁₃×C₄₃
UCS	C₁ ◁ C₂×C₃

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 10062

A minimal faithful permutation representation — 2 generators on 67 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 45);

C2×#23758451 #34117442