Dic₉₁₃₉ #49143164

order	36556
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₁₃₉
unambiguous semidirect product	C₁₉⋊Dic₄₈₁

Splittings

derived, Hall	C₁₃×C₁₉×C₃₇ ⋊ C₄
Hall	C₁₉×C₃₇ ⋊ Dic₁₃
Hall	C₁₃×C₃₇ ⋊ Dic₁₉
Hall	C₁₃×C₁₉ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₂₄₇
Sylow	C₁₉ ⋊ Dic₄₈₁
Sylow	C₁₃ ⋊ Dic₇₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₁₃₉
commutator, hypocenter	C₁₃×C₁₉×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₁₉×C₃₇	C₂
LCS	G ▷ C₁₃×C₁₉×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 36556

A minimal faithful permutation representation — 2 generators on 73 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,13,19,37,2|a^2=e,b^a=b^12,c^a=c^18,d^a=d^36

Conjugacy classes & character table

9142 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36556, 8);

Dic9139 #49143164