Dic₄₇₇₃ #34061840

order	19092
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₇₇₃
unambiguous semidirect product	C₃⋊Dic₁₅₉₁
unambiguous semidirect product	C₄₃⋊Dic₁₁₁

Splittings

derived, Hall	C₃×C₃₇×C₄₃ ⋊ C₄
Hall	C₃₇×C₄₃ ⋊ Dic₃
Hall	C₃×C₄₃ ⋊ Dic₃₇
Hall	C₃×C₃₇ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₁₁
Sylow	C₃₇ ⋊ Dic₁₂₉
Sylow	C₃ ⋊ Dic₁₅₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₇₇₃
commutator, hypocenter	C₃×C₃₇×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₄₃	C₂
LCS	G ▷ C₃×C₃₇×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 19092

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)(6,7)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2,3,37,43,2|a^2=e,b^a=b^2,c^a=c^36,d^a=d^42

Conjugacy classes & character table

4776 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₅₄₆, Dic₁₄₃₁₉
maximal normal subgroup of	Dic₉₅₄₆
maximal quotient of	Dic₁₄₃₁₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19092, 12);

Dic4773 #34061840