Dic₄₇₂ #14273674

order	1888
automorphism	Aut(G) order 438016
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₇₂

Splittings

Sylow

C₅₉ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₄₇₂
commutator, cosocle	C₈×C₅₉	C₂²
Frattini	C₈	C₂×D₅₉
Fitting	C₁₆×C₅₉	C₂
socle, prime radical	C₂×C₅₉	D₈
hypercenter	C₁₆	D₅₉
hypocenter	C₅₉	Q₃₂
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 1888

A minimal faithful permutation representation — 2 generators on 91 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(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:

2^5,59|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^58,c^b=cd,d^b=de,f^b=f^58

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#16837240, #16837241, #16837242, …
maximal normal subgroup of	#16837240, #16837241, #16837242, …
maximal quotient of	#16837207, #16837228, C₂×Dic₄₇₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1888, 93);

Dic472 #14273674