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C₂×#4323 #66984

order	352
isoclinism	isoclinic class of #4323
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂×D₄×C₁₁ ⋊ C₂
Sylow	C₁₁ ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₂×D₄×C₁₁	C₂
socle	C₂²×C₁₁	D₄
hypercenter	C₂×D₄	D₁₁
hypocenter	C₁₁	C₂×D₈
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622362, #10622369, #10622386, …
maximal normal subgroup of	#10622362, #10622369, #10622386, …
maximal quotient of	C₂×#66871, C₂×#66873, #10622580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 126);