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C₄×Dic₂₀ #64987

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₄×Dic₂₀

Splittings

C₅ ⋊ C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₄	D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×C₈	D₅
hypocenter	C₅	C₄×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597957, #10597963, #10597996, …
maximal normal subgroup of	#10597957, #10597963, #10597996, …
maximal quotient of	C₈×Dic₂₀, #10598849, #10598879, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 325);