Dic₉₂ #67265

order	368
isoclinism	isoclinic class of #67263
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₂

Splittings

Sylow

C₂₃ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₉₂
commutator, cosocle	C₄×C₂₃	C₂²
Frattini	C₄	C₂×D₂₃
Fitting	C₈×C₂₃	C₂
socle, prime radical	C₂×C₂₃	D₄
hypercenter	C₈	D₂₃
hypocenter	C₂₃	Q₁₆
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 368

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)

Polycyclic presentation:

2^4,23|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10624982, Dic₁₈₄, #10625009, …
maximal normal subgroup of	#10624982, Dic₁₈₄, #10625009, …
maximal quotient of	#10624998, #10625000, C₂×Dic₉₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 7);

Dic92 #67265