C₂×C₃⋊#20259713 #49149604

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

Aliases

direct product	C₂×C₃⋊#20259713
unambiguous semidirect product	(C₂×C₃)⋊#20259713

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×C₃⋊(D₇₉⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#20259713
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)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,81)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,85,86,67,68,96)(52,84,69,70,66,78)(55,60,83,72,73,79)(58,92,91,76,77,75)(71,97,89,88,87,94)(74,82,93,90,80,95);(1,2)(3,4,5)(6,7,9,8,11,15,12,16,18,10,13,17,14)(19,20,23,29,37,50,69,46,61,62,45,60,80,49,66,77,79,81,94,88,70,51,38,22,26,31,41,55,74,91,63,33,44,32,43,58,68,87,86,93,95,96,97,85,76,54,28,34,42,56,75,90,72,53,40,25,24,30,39,52,71,89,65,83,92,84,64,82,73,36,47,57,59,78,67,48,35,27,21)

Conjugacy classes & character table

1056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36972, 252);

C2×C3⋊#20259713 #49149604