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C₂²⋊C₄×D₁₁ #66933

order	352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂²⋊C₄×D₁₁

Splittings

Fitting	C₁₁×C₂²⋊C₄ ⋊ C₂
Sylow	C₁₁ ⋊ C₂×C₂²⋊C₄

Subgroups

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

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 88

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)(17,19);(12,13,14,15)(16,17)(18,19)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₁×#99, C₂×C₂²⋊C₄×D₁₁, #10623017, …
maximal normal subgroup of	D₁₁×#99, C₂×C₂²⋊C₄×D₁₁, #10623017, …
maximal quotient of	#10622259, D₁₁×#95, #10622277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 75);