C₂×C₅×D₆₆₇ #27211128

order	13340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₆₆₇
unambiguous semidirect product	(C₂×C₂₃)⋊(C₅×D₂₉)
unambiguous semidirect product	(C₂×C₂₉)⋊(C₅×D₂₃)

Splittings

Fitting, socle	C₂×C₅×C₂₃×C₂₉ ⋊ C₂
derived, Hall	C₂₃×C₂₉ ⋊ C₂²×C₅
Hall	C₅×C₂₃×C₂₉ ⋊ C₂²
Hall	C₅×C₂₉ ⋊ C₂×D₂₃
Hall	C₅×C₂₃ ⋊ C₂×D₂₉
Sylow	C₂₉ ⋊ C₂×C₅×D₂₃
Sylow	C₂₃ ⋊ C₂×C₅×D₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₆₆₇
commutator, cosocle, hypocenter	C₂₃×C₂₉	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₂₃×C₂₉	C₂
LCS	G ▷ C₂₃×C₂₉
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 6670

A minimal faithful permutation representation — 2 generators on 59 points:

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

Conjugacy classes & character table

3350 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13340, 33);

C2×C5×D667 #27211128