Dic₆₈₁₅ #41030682

order	27260
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₈₁₅
unambiguous semidirect product	C₄₇⋊Dic₁₄₅

Splittings

derived, Hall	C₅×C₂₉×C₄₇ ⋊ C₄
Hall	C₂₉×C₄₇ ⋊ Dic₅
Hall	C₅×C₄₇ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₂₃₅
Sylow	C₅ ⋊ Dic₁₃₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₈₁₅
commutator, hypocenter	C₅×C₂₉×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₄₇	C₂
LCS	G ▷ C₅×C₂₉×C₄₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 27260

A minimal faithful permutation representation — 2 generators on 85 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);(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)(40,85)(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:

2,5,29,47,2|a^2=e,b^a=b^4,c^a=c^28,d^a=d^46

Conjugacy classes & character table

6818 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₆₃₀
maximal normal subgroup of	Dic₁₃₆₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27260, 8);

Dic6815 #41030682