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D₄×Dic₁₁ #66987

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

Aliases

D₄×Dic₁₁

Splittings

C₁₁ ⋊ C₄×D₄

Subgroups

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

94 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)(12,13,14,15);(1,2,4,6,8,10,11,9,7,5,3)(17,19);(16,17)(18,19)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622369, #10622370, #10622371, …
maximal normal subgroup of	#10622369, #10622370, #10622371, …
maximal quotient of	C₂²⋊C₄×Dic₁₁, #10622563, #10622564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 129);