#66968

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×SD₁₆ ⋊ C₂
Sylow	C₁₁ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	D₄×D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₁₁×SD₁₆	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4341	C₂
hypercenter	SD₁₆	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

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

Polycyclic presentation:

2^5,11|b^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×#66968, #10623293, #10623299, …
maximal normal subgroup of	C₂×#66968, #10623293, #10623299, …
maximal quotient of	#10622370, #10622372, #10622378, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 110);