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C₂²×C₂²⋊D₁₁ #67045

order	352
isoclinism	isoclinic class of Dic₂₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×C₂²⋊D₁₁

Splittings

Fitting	C₂⁴×C₁₁ ⋊ C₂
Sylow	C₁₁ ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₁
commutator, cosocle	C₂×C₁₁	C₂⁴
Frattini	C₂	C₂³×D₁₁
Fitting	C₂⁴×C₁₁	C₂
socle	C₂³×C₁₁	C₂²
hypercenter	C₂⁴	D₁₁
hypocenter	C₁₁	C₂²×D₄
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#66870, #10622569, #10622570, …
maximal normal subgroup of	C₂×#66870, #10622569, #10622570, …
maximal quotient of	C₂²×Dic₁₁⋊C₄, C₂×#66977, C₂²×(C₂×D₁₁)⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 187);