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Q₈×#261 #88241

order	416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	Q₈×C₁₃ ⋊ C₄
Sylow	C₁₃ ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#261
commutator, socle	C₂×C₁₃	C₂²×C₄
Fitting	Q₈×C₁₃	C₄
prime radical, cosocle	C₂×D₁₃	C₂³
hypercenter	Q₈	#261
hypocenter	C₁₃	C₄×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ Q₈

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×#261, #11719106, Q₁₆×#261, …
maximal normal subgroup of	SD₁₆×#261, #11719106, Q₁₆×#261, …
maximal quotient of	#11719074, #11719078, #11719080, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 208);