Dic₇₆₈₅ #44502287

order	30740
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₆₈₅

Splittings

derived, Hall	C₅×C₂₉×C₅₃ ⋊ C₄
Hall	C₂₉×C₅₃ ⋊ Dic₅
Hall	C₅×C₅₃ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₂₆₅
Sylow	C₅ ⋊ Dic₁₅₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₈₅
commutator, hypocenter	C₅×C₂₉×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₅₃	C₂
LCS	G ▷ C₅×C₂₉×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 30740

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

Polycyclic presentation:

2,5,29,53,2|a^2=e,b^a=b^4,c^a=c^28,d^a=d^52

Conjugacy classes & character table

7688 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₃₇₀
maximal normal subgroup of	Dic₁₅₃₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30740, 8);

Dic7685 #44502287