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C₂×C₄×#168 #65209

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₄×#168

Splittings

C₅ ⋊ C₂×C₄×C₈

Subgroups

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

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598110, #10598144, #10598158, …
maximal normal subgroup of	#10598110, #10598144, #10598158, …
maximal quotient of	C₄²×#168, #10600181, C₄×C₄⋊#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 547);