#66995

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×Q₈×C₁₁ ⋊ C₂
Sylow	C₁₁ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₂×Q₈×C₁₁	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4339	C₂
hypercenter	C₂×Q₈	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

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

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622367, #10622368, #10622674, …
maximal normal subgroup of	#10622367, #10622368, #10622674, …
maximal quotient of	#10622577, #10622579, #10622580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 137);