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C₂²×C₁₁⋊C₈ #66973

order	352
isoclinism	isoclinic class of D₁₁
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₁₁⋊C₈

Splittings

C₁₁ ⋊ C₂²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₁₁
commutator, hypocenter	C₁₁	C₂²×C₈
Frattini	C₄	C₂²×D₁₁
Fitting	C₂²×C₄×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₄
cosocle	C₄×C₁₁	C₂³
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 352

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,15)(16,17)(18,19)(20,21)(22,23);(3,4)(13,14,16,18,20,22,23,21,19,17,15);(1,2)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622162, #10622167, #10622188, …
maximal normal subgroup of	#10622162, #10622167, #10622188, …
maximal quotient of	C₂×C₄×C₁₁⋊C₈, C₂×C₄⋊(C₁₁⋊C₈), #10622538, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 115);