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C₂×#6131 #88253

order	416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₂×Q₈×C₁₃ ⋊ C₂
Sylow	C₁₃ ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁴
Frattini	C₂	C₂³×D₁₃
Fitting	C₂×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	C₂³
hypercenter	C₂×Q₈	D₁₃
hypocenter	C₁₃	C₂×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718412, #11718464, #11718465, …
maximal normal subgroup of	#11718412, #11718464, #11718465, …
maximal quotient of	C₂×#88144, C₂×#88146, C₂×#88147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 220);