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C₂×#4343 #67039

order	352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

Fitting	C₂×Q₈×C₁₁ ⋊ C₂
Sylow	C₁₁ ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₁₁
commutator, cosocle	C₂×C₁₁	C₂⁴
Frattini	C₂	C₂³×D₁₁
Fitting	C₂×Q₈×C₁₁	C₂
socle	C₂²×C₁₁	C₂³
hypercenter	C₂×Q₈	D₁₁
hypocenter	C₁₁	C₂×#41
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622364, #10622416, #10622417, …
maximal normal subgroup of	#10622364, #10622416, #10622417, …
maximal quotient of	C₂×#66943, C₂×#66945, C₂×#66946, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 181);