Finite Group Databasebeta

← Prev (#4090) Next (#4092) →

Q₈×Dic₅ #4091

order	160
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)×#1020 (order 7680) · Inn(G) = C₂²×D₅ · Out(G) = C₂³×S₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×Dic₅

Splittings

C₅ ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄
Fitting	C₂×C₅×Q₈	C₂
socle, prime radical, cosocle	C₂²×C₅	C₂³
hypercenter	C₂×Q₈	D₅
hypocenter	C₅	C₄×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 17 points:

(2,3)(4,5)(6,7,8,9);(1,2,4,5,3)(10,12,13,17)(11,14,15,16);(10,11,13,15)(12,16,17,14)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈⋊#170, #65077, #65078, …
maximal normal subgroup of	Q₈⋊#170, #65077, #65078, …
maximal quotient of	C₄⋊C₄×Dic₅, #65270, #65274, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 166);