Finite Group Databasebeta

← Prev (#87902) Next (#87904) →

Q₈×C₅⋊D₅ #87903

order	400
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₅

Aliases

Q₈×C₅⋊D₅

Splittings

Fitting	C₅²×Q₈ ⋊ C₂
Sylow	C₅² ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×C₅⋊D₅
commutator, socle, cosocle	C₂×C₅²	C₂³
Fitting	C₅²×Q₈	C₂
prime radical	C₂×C₅⋊D₅	C₂²
hypercenter	Q₈	C₅⋊D₅
hypocenter	C₅²	C₂×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ Q₈

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 3 generators on 18 points:

(10,11)(12,13)(15,16)(17,18);(1,3,4,8)(2,5,6,7)(14,15,17,18,16);(1,2,4,6)(3,7,8,5)(9,10,12,13,11)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11716877, #11716902, #11717058, …
maximal normal subgroup of	#11716877, #11716902, #11717058, …
maximal quotient of	#11717207, #11717208, #11717209, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 197);