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C₂×#4034 #65831

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×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅	C₂³
hypercenter	C₂×C₄⋊C₄	D₅
hypocenter	C₅	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599592, #10599593, #10599596, …
maximal normal subgroup of	#10599592, #10599593, #10599596, …
maximal quotient of	C₂×#64936, C₂×#64935, C₂×#64945, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1169);