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Q₈×Dic₁₅ #91834

order	480
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×Dic₁₅

Splittings

Hall	C₃×C₅ ⋊ C₄×Q₈
Sylow	C₅ ⋊ Q₈×Dic₃
Sylow	C₃ ⋊ Q₈×Dic₅

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₅×Q₈	C₂
socle, prime radical, cosocle	C₂²×C₃×C₅	C₂³
hypercenter	C₂×Q₈	D₁₅
hypocenter	C₃×C₅	C₄×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11749239, #11749240, #11749241, …
maximal normal subgroup of	#11749239, #11749240, #11749241, …
maximal quotient of	C₄⋊C₄×Dic₁₅, #11751081, #11751085, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 910);