Dic₃₆₈ #13092258

order	1472
automorphism	Aut(G) order 259072
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₈

Splittings

Sylow

C₂₃ ⋊ Q₆₄

Subgroups

	subgroup	G/H
center	C₂	D₃₆₈
commutator, cosocle	C₁₆×C₂₃	C₂²
Frattini	C₁₆	C₂×D₂₃
Fitting	C₂₃×C₃₂	C₂
socle, prime radical	C₂×C₂₃	D₁₆
hypercenter	C₃₂	D₂₃
hypocenter	C₂₃	Q₆₄
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 1472

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

Polycyclic presentation:

2^6,23|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^22,c^b=cd,d^b=de,e^b=ef,g^b=g^22

Conjugacy classes & character table

371 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#15671612, #15671613, #15671614, …
maximal normal subgroup of	#15671612, #15671613, #15671614, …
maximal quotient of	#15671579, #15671597, C₂×Dic₃₆₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1472, 390);

Dic368 #13092258