C₃×#17985163 #27214012

order	13509
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃×#17985163

Splittings

Fitting, socle	C₃×C₁₉×C₇₉ ⋊ C₃
derived, Hall	C₁₉×C₇₉ ⋊ C₃²
Sylow	C₇₉ ⋊ C₃×C₁₉⋊C₃
Sylow	C₁₉ ⋊ C₃×C₇₉⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#17985163
commutator, cosocle, hypocenter	C₁₉×C₇₉	C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₇₉	C₃
LCS	G ▷ C₁₉×C₇₉
UCS	C₁ ◁ C₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 4503

A minimal faithful permutation representation — 2 generators on 101 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,50)(28,40,48)(31,43,45)(34,46,42)(36,38,39)(41,53,63)(44,56,61)(47,59,58)(49,54,55)(51,64,52)(57,65,72)(60,68,70)(62,66,67)(69,73,77)(71,74,75)(76,78,79)(81,82,84)(83,86,87)(85,89,88)(90,94,92)(91,95,93)(96,98,97);(1,2,4,8,15,25,38,22,32,45,58,67,53,24,36,49,62,71,76,78,68,43,17,3,6,11,18,28,41,54,33,12,19,29,42,55,37,51,50,63,72,77,79,73,56,27,7,13,21,31,44,57,66,46,20,5,9,16,26,39,52,64,35,48,61,70,75,65,40,14,23,34,47,60,69,74,59,30,10)(80,81,83,87,92,97,94,82,85,90,95,84,88,93,86,91,96,98,89)(99,100,101)

Conjugacy classes & character table

1509 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#22583041, C₃×#22583037, C₃×#22583033, …
maximal normal subgroup of	C₃×#22583041, C₃×#22583037, C₃×#22583033, …
maximal quotient of	C₂×C₃×#17985163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13509, 11);

C3×#17985163 #27214012