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C₄⋊C₄×D₁₁ #66944

order	352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄⋊C₄×D₁₁

Splittings

Fitting	C₁₁×C₄⋊C₄ ⋊ C₂
Sylow	C₁₁ ⋊ C₂×C₄⋊C₄

Subgroups

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

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₁×#102, #10622384, #10622387, …
maximal normal subgroup of	D₁₁×#102, #10622384, #10622387, …
maximal quotient of	#10622260, #10622262, #10622263, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 86);