C₂₃×#10596815 #27211110

order	13340
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂₃×#10596815

Splittings

Fitting, socle, Hall	C₅×C₂₃×C₂₉ ⋊ C₄
derived, Hall	C₅×C₂₉ ⋊ C₄×C₂₃
Hall	C₂₃×C₂₉ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₂₃ ⋊ #1181
Sylow	C₂₉ ⋊ AGL(1,𝔽₅)×C₂₃
Sylow	C₅ ⋊ C₂₃×#1181

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃	#10596815
commutator, hypocenter	C₅×C₂₉	C₄×C₂₃
Frattini	C₁	G
Fitting, socle	C₅×C₂₃×C₂₉	C₄
prime radical	C₂₃×D₁₄₅	C₂
cosocle	D₁₄₅	C₂×C₂₃
LCS	G ▷ C₅×C₂₉
UCS	C₁ ◁ C₂₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 3335

A minimal faithful permutation representation — 2 generators on 57 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13340, 15);

C23×#10596815 #27211110