C₂×C₅×D₁₃₃₃ #40998638

order	26660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₁₃₃₃
unambiguous semidirect product	(C₂×C₄₃)⋊(C₅×D₃₁)

Splittings

Fitting, socle	C₂×C₅×C₃₁×C₄₃ ⋊ C₂
derived, Hall	C₃₁×C₄₃ ⋊ C₂²×C₅
Hall	C₅×C₃₁×C₄₃ ⋊ C₂²
Hall	C₅×C₄₃ ⋊ C₂×D₃₁
Hall	C₅×C₃₁ ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×C₅×D₃₁
Sylow	C₃₁ ⋊ C₂×C₅×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₃₃₃
commutator, cosocle, hypocenter	C₃₁×C₄₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₃₁×C₄₃	C₂
LCS	G ▷ C₃₁×C₄₃
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13330

A minimal faithful permutation representation — 2 generators on 81 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)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76,77,78,79);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46)(80,81)

Conjugacy classes & character table

6680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26660, 32);

C2×C5×D1333 #40998638