Dic₈₄₃₇ #46798147

order	33748
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₄₃₇
unambiguous semidirect product	C₁₁⋊Dic₇₆₇
unambiguous semidirect product	C₅₉⋊Dic₁₄₃

Splittings

derived, Hall	C₁₁×C₁₃×C₅₉ ⋊ C₄
Hall	C₁₃×C₅₉ ⋊ Dic₁₁
Hall	C₁₁×C₅₉ ⋊ Dic₁₃
Hall	C₁₁×C₁₃ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₄₃
Sylow	C₁₃ ⋊ Dic₆₄₉
Sylow	C₁₁ ⋊ Dic₇₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₄₃₇
commutator, hypocenter	C₁₁×C₁₃×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃×C₅₉	C₂
LCS	G ▷ C₁₁×C₁₃×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 33748

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

Polycyclic presentation:

2,11,13,59,2|a^2=e,b^a=b^10,c^a=c^12,d^a=d^58

Conjugacy classes & character table

8440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33748, 8);

Dic8437 #46798147