C₂×D₃×#15690493 #49149606

order	36972
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×#15690493
unambiguous semidirect product	C₃⋊(C₂²×#15690493)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×#15690493)

Splittings

Fitting, socle	C₂×C₃×C₁₃×C₇₉ ⋊ C₂×C₃
derived	C₃×C₁₃×C₇₉ ⋊ C₂²×C₃
Hall	C₃×#15690493 ⋊ C₂²
Hall	C₁₃×C₇₉ ⋊ C₂×C₃×D₃
Sylow	C₇₉ ⋊ C₂×D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×D₃×C₇₉⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×#15690493
commutator, cosocle, hypocenter	C₃×C₁₃×C₇₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃×C₇₉	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₇₉
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 6162

A minimal faithful permutation representation — 2 generators on 97 points:

(4,5)(7,8,10)(9,12,15)(11,14,13)(16,18,17)(20,21,23)(22,25,30)(24,28,27)(26,32,40)(29,35,37)(31,38,34)(33,42,53)(36,45,50)(39,48,47)(41,51,44)(43,55,68)(46,58,66)(49,61,63)(52,64,60)(54,56,57)(59,71,81)(62,74,79)(65,77,76)(67,72,73)(69,82,70)(75,83,90)(78,86,88)(80,84,85)(87,91,95)(89,92,93)(94,96,97);(1,2)(3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)(19,20,22,26,33,43,56,40,50,63,76,85,71,42,54,67,80,89,94,96,86,61,35,21,24,29,36,46,59,72,51,30,37,47,60,73,55,69,68,81,90,95,97,91,74,45,25,31,39,49,62,75,84,64,38,23,27,34,44,57,70,82,53,66,79,88,93,83,58,32,41,52,65,78,87,92,77,48,28)

Conjugacy classes & character table

2070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36972, 254);

C2×D3×#15690493 #49149606