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Q₈×C₁₃⋊C₃ #64608

order	312
isoclinism	isoclinic class of D₄×C₁₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₁₃⋊C₃

Splittings

Fitting, Hall	Q₈×C₁₃ ⋊ C₃
Sylow	C₁₃ ⋊ C₃×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×C₁₃⋊C₃
commutator, socle, cosocle	C₂×C₁₃	C₂²×C₃
Fitting	Q₈×C₁₃	C₃
prime radical	C₂×C₁₃⋊C₃	C₂²
hypercenter	Q₈	C₁₃⋊C₃
hypocenter	C₁₃	C₃×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ Q₈

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6106⋊C₃, #6107⋊C₃, SD₁₆×C₁₃⋊C₃, …
maximal normal subgroup of	#6106⋊C₃, #6107⋊C₃, SD₁₆×C₁₃⋊C₃, …
maximal quotient of	C₄⋊C₄×C₁₃⋊C₃, C₂×Q₈×C₁₃⋊C₃, Q₈×C₁₃⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 24);