C₂×C₅×D₁₆₄₃ #45659595

order	32860
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₁₆₄₃
unambiguous semidirect product	(C₂×C₅₃)⋊(C₅×D₃₁)

Splittings

Fitting, socle	C₂×C₅×C₃₁×C₅₃ ⋊ C₂
derived, Hall	C₃₁×C₅₃ ⋊ C₂²×C₅
Hall	C₅×C₃₁×C₅₃ ⋊ C₂²
Hall	C₅×C₅₃ ⋊ C₂×D₃₁
Hall	C₅×C₃₁ ⋊ C₂×D₅₃
Sylow	C₅₃ ⋊ C₂×C₅×D₃₁
Sylow	C₃₁ ⋊ C₂×C₅×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₆₄₃
commutator, cosocle, hypocenter	C₃₁×C₅₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₃₁×C₅₃	C₂
LCS	G ▷ C₃₁×C₅₃
UCS	C₁ ◁ C₂×C₅

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 16430

A minimal faithful permutation representation — 2 generators on 91 points:

(3,4,5,6,7)(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)(40,41)(42,43)(44,45)(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)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91);(1,2)(8,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10)(39,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41)

Conjugacy classes & character table

8230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32860, 42);

C2×C5×D1643 #45659595