C₅×C₃⋊(C₂₃⋊Dic₂₃) #44540701

order	31740
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₂₃

Aliases

direct product	C₅×C₃⋊(C₂₃⋊Dic₂₃)
unambiguous semidirect product	C₃⋊(C₅×C₂₃⋊Dic₂₃)
unambiguous semidirect product	C₂₃⋊(C₅×Dic₆₉)

Splittings

derived, Hall	C₃×C₂₃² ⋊ C₄×C₅
Hall	C₃×C₅×C₂₃² ⋊ C₄
Hall	C₅×C₂₃² ⋊ Dic₃
Sylow	C₂₃² ⋊ C₅×Dic₃
Hall	C₃×C₅ ⋊ C₂₃⋊Dic₂₃
Sylow	C₃ ⋊ C₅×C₂₃⋊Dic₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊(C₂₃⋊D₂₃)
commutator, hypocenter	C₃×C₂₃²	C₄×C₅
Frattini	C₂	C₅×C₃⋊(C₂₃⋊D₂₃)
Fitting, socle, prime radical	C₂×C₃×C₅×C₂₃²	C₂
cosocle	C₂×C₃×C₂₃²	C₂×C₅
LCS	G ▷ C₃×C₂₃²
UCS	C₁ ◁ C₂×C₅

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 31740

A minimal faithful permutation representation — 3 generators on 58 points:

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

Conjugacy classes & character table

7950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 39);

C5×C3⋊(C23⋊Dic23) #44540701