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C₄⋊(C₁₁⋊C₈) #66868

order	352
isoclinism	isoclinic class of Dic₂₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

C₄⋊(C₁₁⋊C₈)

Splittings

C₁₁ ⋊ C₄⋊C₈

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₁₁
commutator	C₂×C₁₁	C₂×C₈
Fitting	C₄²×C₁₁	C₂
socle, prime radical	C₂²×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂²
hypercenter	C₄²	D₁₁
hypocenter	C₁₁	C₄⋊C₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 23 points:

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

Polycyclic presentation:

2^5,11|a^2=c,b^2=d,c^2=e,b^a=bd,f^a=f^10

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622094, #10622095, #10622096, …
maximal normal subgroup of	#10622094, #10622095, #10622096, …
maximal quotient of	#10622095, #10622096, C₄⋊(C₁₁⋊C₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 10);