Dic₈₈ #66865

order	352
isoclinism	isoclinic class of D₁₇₆
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₈

Splittings

Sylow

C₁₁ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₈₈
commutator, cosocle	C₈×C₁₁	C₂²
Frattini	C₈	C₂×D₁₁
Fitting	C₁₁×C₁₆	C₂
socle, prime radical	C₂×C₁₁	D₈
hypercenter	C₁₆	D₁₁
hypocenter	C₁₁	Q₃₂
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 352

A minimal faithful permutation representation — 2 generators on 43 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);(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)(34,43)(35,42)(36,41)(37,40)(38,39)

Polycyclic presentation:

2^5,11|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622087, Dic₁₇₆, #10622158, …
maximal normal subgroup of	#10622087, Dic₁₇₆, #10622158, …
maximal quotient of	#10622141, #10622142, C₂×Dic₈₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 7);

Dic88 #66865