C₃×#27131466 #49138786

order	36414
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃×C₁₇

Aliases

direct product

C₃×#27131466

Splittings

Fitting, socle	C₃×C₇×C₁₇² ⋊ C₂×C₃
derived, Hall	C₇×C₁₇² ⋊ C₂×C₃²
Hall	C₃×C₇⋊(C₁₇²⋊C₃) ⋊ C₂
Hall	C₇⋊(C₁₇⋊D₁₇) ⋊ C₃²
Sylow	C₁₇² ⋊ C₃×AGL(1,𝔽₇)
Sylow	C₇ ⋊ C₃×(C₁₇⋊D₁₇)⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#27131466
commutator, cosocle, hypocenter	C₇×C₁₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₇×C₁₇²
UCS	C₁ ◁ C₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 2 generators on 61 points:

(5,8,9,6,7,10)(11,12,14)(13,25,28,17,18,38)(15,36,37,22,23,51)(16,21,32,26,20,34)(19,46,47,29,30,55)(24,39,59,40,41,57)(27,35,52,44,33,54)(31,49,60,50,42,58)(43,45,61,48,56,53);(1,2,3)(4,5,7,10,9,8,6)(11,13,19,31,37,34,53,52,59,57,54,61,32,51,50,29,17)(12,15,24,16,25,42,27,43,46,30,48,44,49,18,26,40,22)(14,20,33,47,58,39,23,38,56,45,28,36,41,60,55,35,21)

Conjugacy classes & character table

1029 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36414, 81);

C3×#27131466 #49138786