Dic₉₂₆₉ #49150504

order	37076
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₂₆₉
unambiguous semidirect product	C₂₃⋊Dic₄₀₃
unambiguous semidirect product	C₃₁⋊Dic₂₉₉

Splittings

derived, Hall	C₁₃×C₂₃×C₃₁ ⋊ C₄
Hall	C₂₃×C₃₁ ⋊ Dic₁₃
Hall	C₁₃×C₃₁ ⋊ Dic₂₃
Hall	C₁₃×C₂₃ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₂₉₉
Sylow	C₂₃ ⋊ Dic₄₀₃
Sylow	C₁₃ ⋊ Dic₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₂₆₉
commutator, hypocenter	C₁₃×C₂₃×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₂₃×C₃₁	C₂
LCS	G ▷ C₁₃×C₂₃×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 37076

A minimal faithful permutation representation — 2 generators on 71 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,13,23,31,2|a^2=e,b^a=b^12,c^a=c^22,d^a=d^30

Conjugacy classes & character table

9272 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37076, 8);

Dic9269 #49150504