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C₂×#4015 #65802

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

C₅ ⋊ C₂×C₄⋊Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
socle, prime radical	C₂³×C₅	C₂³
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599424, #10599425, #10599426, …
maximal normal subgroup of	#10599424, #10599425, #10599426, …
maximal quotient of	C₄×#4015, #10605038, #10605039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1140);