Dic₇₆ #64518

order	304
isoclinism	isoclinic class of #64516
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₆

Splittings

Sylow

C₁₉ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₇₆
commutator, cosocle	C₄×C₁₉	C₂²
Frattini	C₄	C₂×D₁₉
Fitting	C₈×C₁₉	C₂
socle, prime radical	C₂×C₁₉	D₄
hypercenter	C₈	D₁₉
hypocenter	C₁₉	Q₁₆
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Polycyclic presentation:

2^4,19|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^18

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597257, Dic₁₅₂, #10597284, …
maximal normal subgroup of	#10597257, Dic₁₅₂, #10597284, …
maximal quotient of	#10597273, #10597275, C₂×Dic₇₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 7);

Dic76 #64518