#67004

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

Aliases

No aliases for this group.

Splittings

Sylow

C₁₁ ⋊ #137

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting, prime radical	C₁₁×#41	C₂
socle	C₂×C₁₁	C₂×D₄
hypercenter	#41	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

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

Polycyclic presentation:

2^5,11|a^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	#10622438, #10622439, #10622754, …
maximal normal subgroup of	#10622438, #10622439, #10622754, …
maximal quotient of	#10622608, #10622609, #10622617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 146);