#66971

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×Q₁₆ ⋊ C₂
Sylow	C₁₁ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	D₄×D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₁₁×Q₁₆	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4343	C₂
hypercenter	Q₁₆	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

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

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#66971, #10623297, #10623299, …
maximal normal subgroup of	C₂×#66971, #10623297, #10623299, …
maximal quotient of	#10622402, #10622404, #10622408, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 113);