Dic₅₆ #6375

order	224
isoclinism	isoclinic class of D₁₁₂
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,ℤ/16) (order 5376) · Inn(G) = D₅₆ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₆

Splittings

Sylow

C₇ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₅₆
commutator, cosocle	C₇×C₈	C₂²
Frattini	C₈	C₂×D₇
Fitting	C₇×C₁₆	C₂
socle, prime radical	C₂×C₇	D₈
hypercenter	C₁₆	D₇
hypocenter	C₇	Q₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 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);(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)(34,39)(35,38)(36,37)

Polycyclic presentation:

2^5,7|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#89244, Dic₁₁₂, #89315, …
maximal normal subgroup of	#89244, Dic₁₁₂, #89315, …
maximal quotient of	#89298, #89299, C₂×Dic₅₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 7);

Dic56 #6375