Dic₅₁₃₅ #35206707

order	20540
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₃₅
unambiguous semidirect product	C₇₉⋊Dic₆₅

Splittings

derived, Hall	C₅×C₁₃×C₇₉ ⋊ C₄
Hall	C₁₃×C₇₉ ⋊ Dic₅
Hall	C₅×C₇₉ ⋊ Dic₁₃
Sylow	C₇₉ ⋊ Dic₆₅
Hall	C₅×C₁₃ ⋊ Dic₇₉
Sylow	C₁₃ ⋊ Dic₃₉₅
Sylow	C₅ ⋊ Dic₁₀₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₃₅
commutator, hypocenter	C₅×C₁₃×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃×C₇₉	C₂
LCS	G ▷ C₅×C₁₃×C₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 20540

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,2,4)(6,9)(7,8)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2,5,13,79,2|a^2=e,b^a=b^4,c^a=c^12,d^a=d^78

Conjugacy classes & character table

5138 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₂₇₀, Dic₁₅₄₀₅
maximal normal subgroup of	Dic₁₀₂₇₀
maximal quotient of	Dic₁₅₄₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20540, 9);

Dic5135 #35206707