Q₈×Dic₁₀ #65900

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

Q₈×Dic₁₀

C₅ ⋊ Q₈²

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂²×C₅	C₂⁴
Fitting	C₄×C₅×Q₈	C₂
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	Q₈²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

212 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,4,5,3)(14,16,17,21)(15,18,19,20);(2,3)(4,5)(14,15,17,19)(16,20,21,18);(6,8,9,13)(7,10,11,12);(6,7,9,11)(8,12,13,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1238);