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Q₈×Dic₁₃ #88199

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

Aliases

Q₈×Dic₁₃

Splittings

C₁₃ ⋊ C₄×Q₈

Subgroups

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

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 25 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,16,17);(1,2,4,6,8,10,12,13,11,9,7,5,3)(18,20,21,25)(19,22,23,24);(18,19,21,23)(20,24,25,22)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈⋊#1057, #11718449, #11718450, …
maximal normal subgroup of	Q₈⋊#1057, #11718449, #11718450, …
maximal quotient of	C₄⋊C₄×Dic₁₃, #11718642, #11718646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 166);