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C₈×#168 #64673

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₈

Aliases

Splittings

C₅ ⋊ C₈²

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₈	D₅
commutator, hypocenter	C₅	C₈²
Frattini	C₄²	C₂×D₅
Fitting	C₄×C₅×C₈	C₂
socle, prime radical	C₂²×C₅	C₄²
cosocle	C₄²×C₅	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₄×C₈

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×#168, #10597989, #10597990, …
maximal normal subgroup of	C₁₆×#168, #10597989, #10597990, …
maximal quotient of	C₈×#672, #10597928, C₁₆×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 11);