C₁₉⋊C₃×D₁₀₉ #27159090

order	12426
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₉⋊C₃×D₁₀₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₁₀₉ ⋊ C₂×C₃
Hall	C₁₉⋊C₃×C₁₀₉ ⋊ C₂
Hall	C₁₉×D₁₀₉ ⋊ C₃
Sylow	C₁₀₉ ⋊ C₂×C₁₉⋊C₃
Sylow	C₁₉ ⋊ C₃×D₁₀₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₁₀₉	C₂×C₃
LCS	G ▷ C₁₉×C₁₀₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 2071

A minimal faithful permutation representation — 2 generators on 128 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(111,112,114)(113,116,117)(115,119,118)(120,124,122)(121,125,123)(126,128,127);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(110,111,113,117,122,127,124,112,115,120,125,114,118,123,116,121,126,128,119)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	(C₁₉⋊C₃)⋊#89137, D₁₉⋊C₃×D₁₀₉, C₁₉⋊C₃×#89137, …
maximal normal subgroup of	(C₁₉⋊C₃)⋊#89137, D₁₉⋊C₃×D₁₀₉, C₁₉⋊C₃×#89137, …
maximal quotient of	C₁₉⋊C₃×Dic₁₀₉, C₂×C₁₉⋊C₃×D₁₀₉, #4294×D₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12426, 14);

C19⋊C3×D109 #27159090