#66962

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×M₁₆ ⋊ C₂
Sylow	C₁₁ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₄
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₁₁×M₁₆	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4339	C₂
hypercenter	M₁₆	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6,8,10,11,9,7,5,3)(12,15,16,23,17,24,14,21)(13,19,18,22,20,27,25,26);(12,14,17,16)(13,18,20,25)(15,23,24,21)(19,26,27,22);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,18)(15,22)(16,25)(17,20)(19,23)(21,27)(24,26)

Polycyclic presentation:

2^5,11|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^10,c^b=ce

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622359, #10622361, #10622365, …
maximal normal subgroup of	#10622359, #10622361, #10622365, …
maximal quotient of	#10622315, #10622316, #10622323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 104);